<!DOCTYPE html><html lang="en">
      
      
        
    
    
    
    

      
      
      <head>
        <meta charset="utf-8">
        <meta name="format-detection" content="telephone=no">  
        
        <title>Lecture 25: Random Walks | Video Lectures | Mathematics for Computer Science | Electrical Engineering and Computer Science | MIT OpenCourseWare</title>
    <!-- Begin Automatic Metadata Insertion -->
    <meta content="6-042j-mathematics-for-computer-science-fall-2010" name="WT.cg_n">
    <meta content="Lecture 25: Random Walks" name="WT.cg_s">
    <meta content="Discusses random walks and their non-intuitive effect on systems, such as gambling at roulette and gambler's ruin." name="Description">
    <meta content="Leighton, Tom" name="Author">
    <meta content="Dijk, Marten van " name="Author">
    <meta content="6.042J,6.042,18.062J,18.062,random walks,gambler's ruin,roulette,martingale,biased,unbiased,drift,Computer Science,Probability and Statistics,Applied Mathematics,Discrete Mathematics" name="keywords">
    <meta content="6.042J Mathematics for Computer Science | Lecture 25: Random Walks" name="Search_Display">
    <meta content="Computer Science" itemprop="about">
    <meta content="Probability and Statistics" itemprop="about">
    <meta content="Applied Mathematics" itemprop="about">
    <meta content="Discrete Mathematics" itemprop="about">
    <!-- End Automatic Metadata Insertion -->

	<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/grid.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/base.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/menu.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/jquery.bubblepopup.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/search.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/courses.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/courses_new.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/jquery.jscrollpane.css">
<link title="default" rel="stylesheet" type="text/css" href="../../../common/styles/media_tabs.css">
	<link href="../../../common/xml/ocwcc.rdf" type="application/rdf+xml" rel="metadata">
	<link rel="canonical" href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-25-random-walks/">
	<link rel="apple-touch-icon" href="../../../common/images/apple-touch-icon.png">
	
	
      
	
	<script type="text/javascript" src="../../../common/scripts/jquery.js"></script>
	<script type="text/javascript" src="../../../common/scripts/ocw-media-utils-offline.js"></script>
	<script type="text/javascript" src="../../../common/scripts/ocw-offline.js"></script>
	<script type="text/javascript" src="../../../common/scripts/jquery.bubblepopup.min.js"></script>
	<script type="text/javascript" src="../../../common/scripts/jquery-ui.min.js"></script>
	<script type="text/javascript" src="../../../common/scripts/jquery.jscrollpane.min.js"></script>
	<script type="text/javascript" src="../../../common/scripts/expandy.js"></script>
	<script type="text/javascript" src="../../../common/scripts/bubble-popup-offline.js"></script>
	
	
	
    <script type="text/javascript">
      $(document).ready(function() {
        $("#tabs").tabs();
        IpadScroller();
      });
    </script>
    
    
    
    
    
    
      
       
		 

        
        
        

        
        
        
        
        
        
        
        
        
      </head>
    <body itemscope itemtype="http://schema.org/WebPage">
        
	

        <header id="top">
			<div id="grid">
				
				
					
<div id="portletwrapper-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d686561646572" class="portletWrapper kssattr-portlethash-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d686561646572">
<div class="portletStaticText portlet-static-site-header">
<!--googleoff: index-->
<div class="grid_6 alpha" id="banner"><a href="https://ocw.mit.edu/"><img src="../../../common/images/ocw_mast.png" class="logo" alt="MIT OpenCourseWare, Massachusetts Institute of Technology"></a></div>
<div class="grid_6 omega" id="subscribe">
<aside class="module" aria-label="Connect with OCW">
<table class="social">
    <tbody>
        <tr>
            <td class="socialbutton"><a aria-label="Subscribe to the OCW Newsletter" href="https://ocw.mit.edu/subscribe/index.htm?utm_source=header"><img src="../../../common/images/trans.gif" alt="An icon depicting an envelope.">Subscribe to the OCW Newsletter</a></td>
            <td>
<a aria-label="Facebook" href="https://facebook.com/mitocw"><img src="../../../common/images/icon_fb.png" alt="Click to visit our Facebook page."></a>  <a aria-label="Instagram" href="https://www.instagram.com/mitocw/"><img src="https://ocw.mit.edu/images/icon_ig.png" alt="Click to visit our Instagram page."></a> <a aria-label="Twitter" href="https://twitter.com/mitocw"><img src="https://ocw.mit.edu/images/icon_twitter.png" alt="Click to visit our Twitter feed."></a><a aria-label="YouTube" href="https://www.youtube.com/mitocw" style="font-size: 12.208px;"><img src="https://ocw.mit.edu/images/icon_yt.png" alt="Click to visit our YouTube page."></a><span style="font-size: 12.208px;">   </span><a aria-label="Open Matters blog on WordPress" href="https://www.ocw-openmatters.org/" style="font-size: 12.208px;"><img src="https://ocw.mit.edu/images/icon_wp.png" alt="Click to visit our blog on WordPress."></a>
</td>
        </tr>
    </tbody>
</table>
</aside><nav aria-label="Help Links" class="helplinks">     <a aria-label="OCW Site Help" href="https://ocw.mit.edu/help">Help</a><span aria-hidden="true">|</span>     <a href="../../../common/about/contactus.htm">Contact Us</a>   </nav>
</div>
<div class="clear"> </div>
<!--googleon: index-->
</div>

</div>

<div id="portletwrapper-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a6d6567612d6d656e75" class="portletWrapper kssattr-portlethash-6f63772e746f70706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a6d6567612d6d656e75">
<div class="portletStaticText portlet-static-mega-menu"><div><nav id="mega" class="grid_8 alpha" aria-label="Site">
<ul id="menu" role="presentation">
    <li id="menu_home">
<a href="https://ocw.mit.edu/" aria-label="Homepage"><img src="../../../common/images/top-nav_home.png" class="home_icon" alt="Click for site home page."></a><!-- Begin Home Item -->
</li>
    <!-- End Home Item -->
    <li id="drop_1" aria-label="Find Courses">
<a href="#" aria-hidden="true">FIND COURSES</a><!-- Begin 5 columns Item -->
    <div class="dropdown_5columns-a mega-courses">
    <div class="col_1a">
    <div class="row_1a">
<nav aria-labelledby="mm-find-courses-by">     <span id="mm-find-courses-by" class="nav" aria-hidden="true">Find courses by:</span>
    <ul class="find_by" role="presentation">
        <li><a href="https://ocw.mit.edu/courses/find-by-topic/">Topic</a></li>
        <li><a href="https://ocw.mit.edu/courses/find-by-number/">MIT Course Number</a></li>
        <li><a href="https://ocw.mit.edu/courses/find-by-department/">Department</a></li>
    </ul>
    </nav>     <nav aria-labelledby="mm-collections">     <span id="mm-collections" class="nav" aria-hidden="true">Collections</span>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/new-courses/">New Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/most-visited-courses/">Most Visited Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/ocw-scholar/">OCW Scholar Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/audio-video-courses/">Audio/Video Lectures</a></li>
        <li><a href="https://ocw.mit.edu/courses/online-textbooks/">Online Textbooks</a></li>
        <li><a href="https://ocw.mit.edu/resources/">Supplemental Resources</a></li>
        <li><a href="https://ocw.mit.edu/high-school/">OCW Highlights for High School</a></li>
        <li><a href="https://ocw.mit.edu/courses/mitx-related-courseware/">MITx &amp; Related OCW Courses</a></li>
        <li><a href="https://ocw.mit.edu/courses/mit-open-learning-library/">MIT Open Learning Library</a></li>
    </ul>
    </nav>     <nav class="col_1b" aria-labelledby="mm-translated-courses">     <span id="mm-translated-courses" class="nav" aria-hidden="true" style="line-height: 1.3;">Cross-Disciplinary Topic Lists</span>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/energy-courses">Energy</a></li>
        <li><a href="https://ocw.mit.edu/courses/entrepreneurship">Entrepreneurship</a></li>
        <li><a href="https://ocw.mit.edu/courses/environment-courses">Environment</a></li>
        <li><a href="https://ocw.mit.edu/courses/intro-programming">Introductory Programming</a></li>
        <li><a href="https://ocw.mit.edu/courses/life-sciences">Life Sciences</a></li>
        <li><a href="https://ocw.mit.edu/courses/transportation-courses">Transportation</a></li>
    </ul>
    </nav>
</div>
    <div class="row_1b"><nav aria-labelledby="mm-cross-disciplinary-topic-lists">     <span id="mm-cross-disciplinary-topic-lists" class="nav" aria-hidden="true">Translated Courses</span>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/translated-courses/traditional-chinese" aria-label="Traditional Chinese">繁體字 / Traditional Chinese</a></li>
        <li><a href="https://ocw.mit.edu/courses/translated-courses/turkish" aria-label="Turkish">Türkçe / Turkish</a></li>
        <li><a href="https://ocw.mit.edu/courses/translated-courses/korean" aria-label="Korean">(비디오)한국 / Korean</a></li>
    </ul>
    </nav></div>
    </div>
    </div>
    </li>
    <li id="drop_2">
<a href="#" aria-label="For Educators">For Educators</a>
    <div class="dropdown_1column-a" style="width: 270px;"><nav aria-labelledby="mm-find-courses-by">
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/educator/chalk-radio-podcast">Chalk Radio Podcast</a></li>
        <li><a href="https://ocw.mit.edu/educator/">OCW Educator Portal</a></li>
    </ul>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/courses/instructor-insights/">Instructor Insights by Department</a></li>
        <li><a href="https://openlearning.mit.edu/campus/digital-innovations/">Residential Digital Innovations </a></li>
    </ul>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/high-school/">OCW Highlights for High School</a></li>
    </ul>
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/educator/additional-resources/">Additional Resources</a></li>
    </ul>
    </nav></div>
    </li>
    <li id="drop_3">
<a href="#" aria-hidden="true">Give Now</a>
    <div class="dropdown_1column-a"><nav class="col_1" aria-label="Donate">
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/give/">Make a Donation</a></li>
        <li><a href="https://ocw.mit.edu/give/why-give/">Why Give?</a></li>
        <li><a href="https://ocw.mit.edu/give/our-supporters/">Our Supporters</a></li>
        <li><a href="https://ocw.mit.edu/give/other-ways-to-contribute/">Other Ways to Contribute</a></li>
        <li><a href="https://ocw.mit.edu/support/">Become a Corporate Sponsor</a></li>
    </ul>
    </nav></div>
    </li>
    <li id="drop_4">
<a href="#" aria-hidden="true">About</a>
    <div class="dropdown_1column-a"><nav class="col_1" aria-label="About">
    <ul role="presentation">
        <li><a href="https://ocw.mit.edu/about/">About MIT OpenCourseWare</a></li>
        <li><a href="https://ocw.mit.edu/about/site-statistics/">Site Statistics</a></li>
        <li><a href="https://ocw.mit.edu/about/ocw-stories/">OCW Stories</a></li>
        <li><a href="https://ocw.mit.edu/about/newsletter/">Newsletter</a></li>
        <li><a href="https://chalk-radio.simplecast.com/">Chalk Radio Podcast</a></li>
        <li><a href="https://www.ocw-openmatters.org/">Open Matters Blog</a></li>
    </ul>
    </nav></div>
    </li>
</ul>
</nav></div></div>

</div>





<!--googleoff: index-->
<script>
  (function() {
	var cx = '012626166551961672889:owjdpuboktq';
	var gcse = document.createElement('script');
	gcse.type = 'text/javascript';
	gcse.async = true;
	gcse.src = 'https://cse.google.com/cse.js?cx=' + cx;
	var s = document.getElementsByTagName('script')[0];
	s.parentNode.insertBefore(gcse, s);
  })();
  window.onload = function(){
	document.getElementById('gsc-i-id1').placeholder = 'Search';	
  };

$(document).ready(function(){

$('.advanceSearch a').keydown(function(event){showSearchTips($(this),event);})
$('#searchTipsModal').keydown(function(event){showSearchTips($(this),event);})

function showSearchTips(obj,evt) {
// if pressed enter key
	if ( evt.which == 13 || evt.which == 32) {
		showModal();
		ga('send', 'pageview', "AdvanceSearch");
		$(".advanceSearch a").attr('aria-expanded', 'true');
		$('.searchTipsModal').focus();
		evt.preventDefault();
		}
	if ( evt.which == 27 ) {
		hideModal();
		$(".advanceSearch a").attr('aria-expanded', 'false');
		var modal = document.getElementById('searchTipsModal');
		modal.style.display = "none";
		$(".advanceSearch a").focus();
		evt.preventDefault();
		}
}

});
function showModal(){
	var modal = document.getElementById('searchTipsModal');
	modal.style.display = "block";
	ga('send', 'pageview', "AdvanceSearch");
	document.getElementById("searchTipsBtn").setAttribute("aria-expanded", true);
	window.setTimeout(function () {
    document.getElementById('searchTipsModal').focus(); }, 0);
}

function hideModal(){
	var modal = document.getElementById('searchTipsModal');
	modal.style.display = "none";
	document.getElementById("searchTipsBtn").setAttribute("aria-expanded", false);
}
</script>
<div id="search" role="search" class="grid_4 omega">
    	<table class="search">
				<tbody>
					<tr>
						<td><div class="searchboxheader"><searchbox-only resultsurl="/search/ocwsearch.htm"></searchbox-only></div></td>
						<td>
							<div class="advanceSearch">
								<a id="searchTipsBtn" onclick="showModal();" href="#" role="button" aria-label="search tips" aria-expanded="false" aria-describedby="searchtips">Search Tips</a>

								<!-- The Modal -->
								<div id="searchTipsModal" class="modal" tabindex="-1">
								  <!-- Modal content -->
									<div class="modal-content">
										<div class="modal-body">
											<button class="close" onclick="hideModal();" aria-label="close">X</button>
											<span>
												<b>Exclude words from your search</b>
												<br>Put - in front of a word you want to leave out. For example, jaguar speed -car
												<br><br>
												<b>Search for an exact match</b>
												<br>Put a word or phrase inside quotes. For example, "tallest building".
												<br><br>
												<b>Search for wildcards or unknown words</b>
												<br>Put a * in your word or phrase where you want to leave a placeholder. For example, "largest * in the world".
												<br><br>
												<b>Search within a range of numbers</b>
												<br>Put .. between two numbers. For example, camera $50..$100.
												<br><br>
												<b>Combine searches</b>
												<br>Put "OR" between each search query. For example,  marathon OR race.
												<br><br>
											</span>
										</div>
								  </div>
								</div>
							</div>
						</td>
					</tr>
				</tbody>
		</table>
</div>
<div class="clear"></div>
<!--googleon: index-->
<!-- *end header* -->

				
				
			</div>
<!-- top grid end -->
		</header><!-- top end -->
			
		<div id="center_media">
      	<div id="grid">
      		<div id="left">
        		<nav id="breadcrumb_media" aria-label="Breadcrumb">
                	<p>

    <a href="https://ocw.mit.edu/">Home</a>
    
        »
        
    
    
        
            <a href="https://ocw.mit.edu/courses">Courses</a>
            
                »
                
            
            
         
    
    
        
            <a href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science">Electrical Engineering and Computer Science</a>
            
                »
                
            
            
         
    
    
        
            <a href="../../../contents/index.htm">Mathematics for Computer Science</a>
            
                »
                
            
            
         
    
    
        
            <a href="../../../contents/video-lectures/index.htm">Video Lectures</a>
            
                »
                
            
            
         
    
    
        
            
            
            Lecture 25: Random Walks
         
    
</p>

            	</nav>
            	<div class="clear"></div>
        		<div id="media_title">
        		<h1 class="title" itemprop="name" property="dct:title">
        <span class="" id="parent-fieldname-title">
            Lecture 25: Random Walks
        </span>
    </h1>
        		</div>
           		<div class="clear"></div>
           		<div id="course_wrapper_media">
           			<nav id="course_nav" aria-label="Course">
           				<script language="javascript" type="text/javascript">
function toggleMenu(objID) {
  if (!document.getElementById) return;
  var ob = document.getElementById(objID);
  ob.className = (ob.className == 'selected')?'': 'selected';
}
function toggleClass(id)
{
  var divtoggleClass= document.getElementById(id);
  divtoggleClass.className = (divtoggleClass.className == 'mO')?'mC': 'mO';
  return false;
}
function changeAlt(id)
{
  id.alt = (id.alt == 'Expand Menu')?'Collapse Menu' : 'Expand Menu';
  id.title = (id.title == 'Expand Menu')?'Collapse Menu' : 'Expand Menu';
}
</script>
<!--Left Nav Starts -->


	<ul>			  
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/index.htm">
		                  Course Home  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/syllabus/index.htm">
		                  Syllabus  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/calendar/index.htm">
		                  Calendar  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/readings/index.htm">
		                  Readings  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="selected">
			   			<a href="../../../contents/video-lectures/index.htm">
		                  Video Lectures  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/recitations/index.htm">
		                  Recitations  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/assignments/index.htm">
		                  Assignments  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    <li class="">
			   			<a href="../../../contents/exams/index.htm">
		                  Exams  			                
	                    </a>
		        </li>
		    
         	
	
	
	    	
	    	    
		    
         	
	<!--second tal block close-->  
	
</ul>


<!--Left Nav Ends -->





           			</nav>
           			<main id="course_inner_media" aria-labelledby="media_title">
      					 
        <div class="" id="parent-fieldname-text">
            
            
        </div>
    
      					     
    
    



<script type="text/javascript">var caption_embed_1 ={'English - US': '/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-25-random-walks/56iFMY8QW2k.srt'}</script>     
     <div id="media-embed">
         <div class="attention_message" id="embed_1">
<p>Flash and JavaScript are required for this feature.</p>
<p>Download the video from <a href="http://itunes.apple.com/us/itunes-u/lecture-25-random-walks/id503873536?i=110644958">iTunes U</a> or the <a href="http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec25_300k.mp4">Internet Archive</a>.</p>
</div>
     </div>
    
     <script type="text/javascript">ocw_embed_chapter_media('embed_1', 'https://www.youtube.com/v/56iFMY8QW2k', 'youtube', '/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-25-random-walks', 'https://img.youtube.com/vi/56iFMY8QW2k/0.jpg',0,0, 'https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-25-random-walks/56iFMY8QW2k.srt')</script>
	 
	 
	 		
	 		<div id="transcript1"></div>
				 <script type="text/javascript">createThreePlayParams(2, 781490, "embed_1", 0, 0)</script>
			 
	

     <div id="media_resource_next_prev_nav" style="margin-top: 1em;">
        <p>
        
            <a href="../../../contents/video-lectures/lecture-24-large-deviations/index.htm">
                <img src="../../../common/images/btn_previous_resource.png" style="margin: 0 30px 0 50px;" alt="Previous track" title="Previous track"></a>
     	
     	
        
       </p>
     </div>
 


<script type="text/javascript">
		window.onload=function(){
		init();
		
		}
		var tabLinks = new Array();
		var contentDivs = new Array();
		function init() {
		  // Grab the tab links and content divs from the page
		  var tabListItems = document.getElementById('tabs').childNodes;
		  for ( var i = 0; i < tabListItems.length; i++ ) {
			if ( tabListItems[i].nodeName == "LI" ) {
			  var tabLink = getFirstChildWithTagName( tabListItems[i], 'A' );
			  var id = getHash( tabLink.getAttribute('href') );
			  tabLinks[id] = tabLink;
			  contentDivs[id] = document.getElementById( id );
			}
		  }
		  // Assign onclick events to the tab links, and
		  // highlight the first tab
		  var i = 0;
		  for ( var id in tabLinks ) {
			tabLinks[id].onclick = showTab;
			tabLinks[id].onfocus = function() { this.blur() };
			if ( i == 0 ) tabLinks[id].className = 'selected';
			i++;
		  }
		  // Hide all content divs except the first
		  var i = 0;
		  for ( var id in contentDivs ) {
			if ( i != 0 ) contentDivs[id].className = 'tabContent hide';
			i++;
		  }
		}
		function showTab() {
		  var selectedId = getHash( this.getAttribute('href') );
		  // Highlight the selected tab, and dim all others.
		  // Also show the selected content div, and hide all others.
		  for ( var id in contentDivs ) {
			if ( id == selectedId ) {
			  tabLinks[id].className = 'selected';
			  contentDivs[id].className = 'tabContent';
			} else {
			  tabLinks[id].className = '';
			  contentDivs[id].className = 'tabContent hide';
			}
		  }
		  // Stop the browser following the link
		  return false;
		}
		function getFirstChildWithTagName( element, tagName ) {
		  for ( var i = 0; i < element.childNodes.length; i++ ) {
			if ( element.childNodes[i].nodeName == tagName ) return element.childNodes[i];
		  }
		}
		function getHash( url ) {
		  var hashPos = url.lastIndexOf ( '#' );
		  return url.substring( hashPos + 1 );
		}
 </script>	
 

  <div id="media_tabs">
     
        <ul id="tabs">
            <li class="first">
                <a href="#vid_about" class="selected">About this Video</a>
            </li>
            <li class="">
                <a href="#vid_index" class="">Playlist</a>
            </li>
            <li class="">
                <a href="#vid_playlist" class="">Transcript</a>
            </li>
            <li class="">
                <a href="#vid_related" class="">Download this Video</a>
            </li>
        </ul>
   
        <div id="vid_about" itemprop="description" class="tabContent">
<p><strong>Description:</strong> Discusses random walks and their non–intuitive effect on systems, such as gambling at roulette and gambler's ruin.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
</div>
        <div id="vid_index" itemprop="description" class="tabContent hide">
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-1-introduction-and-proofs/index.htm">
<img src="https://img.youtube.com/vi/L3LMbpZIKhQ/default.jpg" title="Lecture 1: Introduction and Proofs" alt="Lecture 1: Introduction and Proofs">
<p>Lecture 1: Introduction and...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-2-induction/index.htm">
<img src="https://img.youtube.com/vi/z8HKWUWS-lA/default.jpg" title="Lecture 2: Induction" alt="Lecture 2: Induction">
<p>Lecture 2: Induction</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-3-strong-induction/index.htm">
<img src="https://img.youtube.com/vi/NuGDkmwEObM/default.jpg" title="Lecture 3: Strong Induction" alt="Lecture 3: Strong Induction">
<p>Lecture 3: Strong Induction</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-4-number-theory-i/index.htm">
<img src="https://img.youtube.com/vi/NuY7szYSXSw/default.jpg" title="Lecture 4: Number Theory I" alt="Lecture 4: Number Theory I">
<p>Lecture 4: Number Theory I</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-5-number-theory-ii/index.htm">
<img src="https://img.youtube.com/vi/XX7ePR21Ook/default.jpg" title="Lecture 5: Number Theory II" alt="Lecture 5: Number Theory II">
<p>Lecture 5: Number Theory II</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-6-graph-theory-and-coloring/index.htm">
<img src="https://img.youtube.com/vi/h9wxtqoa1jY/default.jpg" title="Lecture 6: Graph Theory and Coloring" alt="Lecture 6: Graph Theory and Coloring">
<p>Lecture 6: Graph Theory and...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-7-matching-problems/index.htm">
<img src="https://img.youtube.com/vi/5RSMLgy06Ew/default.jpg" title="Lecture 7: Matching Problems" alt="Lecture 7: Matching Problems">
<p>Lecture 7: Matching Problems</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-8-graph-theory-ii-minimum-spanning-trees/index.htm">
<img src="https://img.youtube.com/vi/GJpt_3ie4WU/default.jpg" title="Lecture 8: Graph Theory II: Minimum Spanning Trees" alt="Lecture 8: Graph Theory II: Minimum Spanning Trees">
<p>Lecture 8: Graph Theory II:...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-9-communication-networks/index.htm">
<img src="https://img.youtube.com/vi/bTyxpoi2dmM/default.jpg" title="Lecture 9: Communication Networks" alt="Lecture 9: Communication Networks">
<p>Lecture 9: Communication Ne...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-10-graph-theory-iii/index.htm">
<img src="https://img.youtube.com/vi/DOIp5D7VMS4/default.jpg" title="Lecture 10: Graph Theory III" alt="Lecture 10: Graph Theory III">
<p>Lecture 10: Graph Theory III</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-11-relations-partial-orders-and-scheduling/index.htm">
<img src="https://img.youtube.com/vi/1nScXLQAQ9A/default.jpg" title="Lecture 11: Relations, Partial Orders, and Scheduling" alt="Lecture 11: Relations, Partial Orders, and Scheduling">
<p>Lecture 11: Relations, Part...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-12-sums/index.htm">
<img src="https://img.youtube.com/vi/fAeShezAGLE/default.jpg" title="Lecture 12: Sums" alt="Lecture 12: Sums">
<p>Lecture 12: Sums</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-13-sums-and-asymptotics/index.htm">
<img src="https://img.youtube.com/vi/X9eErxRjQEI/default.jpg" title="Lecture 13: Sums and Asymptotics" alt="Lecture 13: Sums and Asymptotics">
<p>Lecture 13: Sums and Asympt...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-14-divide-and-conquer-recurrences/index.htm">
<img src="https://img.youtube.com/vi/Kqf0uO0oV6s/default.jpg" title="Lecture 14: Divide and Conquer Recurrences" alt="Lecture 14: Divide and Conquer Recurrences">
<p>Lecture 14: Divide and Conq...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-15-linear-recurrences/index.htm">
<img src="https://img.youtube.com/vi/TWBB-JlmYUc/default.jpg" title="Lecture 15: Linear Recurrences" alt="Lecture 15: Linear Recurrences">
<p>Lecture 15: Linear Recurrences</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-16-counting-rules-i/index.htm">
<img src="https://img.youtube.com/vi/pNt5Ll6hGqo/default.jpg" title="Lecture 16: Counting Rules I" alt="Lecture 16: Counting Rules I">
<p>Lecture 16: Counting Rules I</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-17-counting-rules-ii/index.htm">
<img src="https://img.youtube.com/vi/09yIb3VHhMI/default.jpg" title="Lecture 17: Counting Rules II" alt="Lecture 17: Counting Rules II">
<p>Lecture 17: Counting Rules II</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-18-probability-introduction/index.htm">
<img src="https://img.youtube.com/vi/SmFwFdESMHI/default.jpg" title="Lecture 18: Probability Introduction" alt="Lecture 18: Probability Introduction">
<p>Lecture 18: Probability Int...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-19-conditional-probability/index.htm">
<img src="https://img.youtube.com/vi/E6FbvM-FGZ8/default.jpg" title="Lecture 19: Conditional Probability" alt="Lecture 19: Conditional Probability">
<p>Lecture 19: Conditional Pro...</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-20-independence/index.htm">
<img src="https://img.youtube.com/vi/l1BCv3qqW4A/default.jpg" title="Lecture 20: Independence" alt="Lecture 20: Independence">
<p>Lecture 20: Independence</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-21-random-variables/index.htm">
<img src="https://img.youtube.com/vi/MOfhhFaQdjw/default.jpg" title="Lecture 21: Random Variables" alt="Lecture 21: Random Variables">
<p>Lecture 21: Random Variables</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-22-expectation-i/index.htm">
<img src="https://img.youtube.com/vi/gGlMSe7uEkA/default.jpg" title="Lecture 22: Expectation I" alt="Lecture 22: Expectation I">
<p>Lecture 22: Expectation I</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-23-expectation-ii/index.htm">
<img src="https://img.youtube.com/vi/oI9fMUqgfxY/default.jpg" title="Lecture 23: Expectation II" alt="Lecture 23: Expectation II">
<p>Lecture 23: Expectation II</p></a>
</div>
<div class="related-media-thumbnail">
<a href="../../../contents/video-lectures/lecture-24-large-deviations/index.htm">
<img src="https://img.youtube.com/vi/q4mwO2qS2z4/default.jpg" title="Lecture 24: Large Deviations" alt="Lecture 24: Large Deviations">
<p>Lecture 24: Large Deviations</p></a>
</div>
<div class="related-media-thumbnail-nolink">
<div class="now-playing-resource">Now Playing</div>
<img src="https://img.youtube.com/vi/56iFMY8QW2k/default.jpg" title="Lecture 25: Random Walks" alt="Lecture 25: Random Walks">
<p>Lecture 25: Random Walks</p>
</div>
</div>
        <div id="vid_playlist" itemprop="description" class="tabContent hide">
<ul><li><a class="transcript-link" title="Open in a new window." alt="Open in a new window." style="text-decoration: none; font-size: 1.0em;" target="_blank" text-decoration: none font-size: href="../../../contents/video-lectures/lecture-25-random-walks/56iFMY8QW2k.pdf"> Download English-US transcript (PDF)</a></li></ul>
<p><span m="400">The</span> <span m="520">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1660">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2430">Creative</span> <span m="2830">Commons</span> <span m="3240">license.</span> <span m="4340">Your</span> <span m="4530">support</span> <span m="5030">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high-quality</span> <span m="8440">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation,</span> <span m="12000">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13120">materials</span> <span m="13660">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14780">courses,</span> <span m="15900">visit</span> <span m="16110">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="23480">PROFESSOR: Now</span> <span m="23590">today,</span> <span m="24070">we're</span> <span m="24180">going</span> <span m="24270">to</span> <span m="24330">talk</span> <span m="24780">about</span> <span m="25190">random</span> <span m="25650">walks.</span> <span m="26740">And</span> <span m="26960">in</span> <span m="27030">particular,</span> <span m="27580">we're</span> <span m="27680">going</span> <span m="27770">to</span> <span m="27830">look</span> <span m="28060">at</span> <span m="28170">a</span> <span m="28250">classic</span> <span m="28780">phenomenon</span> <span m="29940">known</span> <span m="30300">as</span> <span m="30460">Gamblers</span> <span m="31050">Ruin.</span> <span m="32750">It's</span> <span m="32960">a</span> <span m="33020">great</span> <span m="33290">way</span> <span m="33430">to</span> <span m="33540">end</span> <span m="33780">the</span> <span m="33870">term,</span> <span m="34760">because</span> <span m="35520">the</span> <span m="35650">solution</span> <span m="36190">requires</span> <span m="36720">several</span> <span m="36990">of</span> <span m="37260">the</span> <span m="37370">techniques</span> <span m="38050">that</span> <span m="38170">we've</span> <span m="38370">developed</span> <span m="39390">since</span> <span m="39790">the</span> <span m="39870">midterm.</span> </p>
<p><span m="40740">So</span> <span m="40900">it's</span> <span m="41030">actually</span> <span m="41360">a</span> <span m="41410">good</span> <span m="41660">review.</span> <span m="42500">We'll</span> <span m="42770">review</span> <span m="43150">recurrences.</span> <span m="43730">We'll</span> <span m="44010">review</span> <span m="44340">a</span> <span m="44400">lot</span> <span m="44560">of</span> <span m="44600">probability</span> <span m="45160">laws.</span> <span m="46120">And</span> <span m="47490">it's</span> <span m="47650">actually</span> <span m="47930">a</span> <span m="47960">nice</span> <span m="48200">problem</span> <span m="48550">to</span> <span m="48640">look</span> <span m="48880">at.</span> <span m="49050">It's</span> <span m="49540">another</span> <span m="49910">example</span> <span m="50410">where</span> <span m="50510">you</span> <span m="50600">get</span> <span m="50730">a</span> <span m="50820">non-intuitive</span> <span m="51560">solution</span> <span m="52090">using</span> <span m="52400">probability.</span> <span m="53740">And</span> <span m="53960">if</span> <span m="54050">you</span> <span m="54160">like</span> <span m="54420">to</span> <span m="54530">gamble,</span> <span m="55600">it's</span> <span m="55780">really</span> <span m="56240">good</span> <span m="56560">that</span> <span m="56630">you</span> <span m="56770">look</span> <span m="56950">at</span> <span m="57010">this</span> <span m="57170">problem</span> <span m="57590">before</span> <span m="58050">you</span> <span m="58160">go</span> <span m="58370">to</span> <span m="58450">Vegas</span> <span m="59010">or</span> <span m="59090">down</span> <span m="59330">to</span> <span m="59410">Foxwoods.</span> </p>
<p><span m="61050">Now</span> <span m="61460">the</span> <span m="61590">Gambler's</span> <span m="62000">Ruin</span> <span m="62320">problem,</span> <span m="65950">you</span> <span m="66130">start</span> <span m="66520">with</span> <span m="66650">n</span> <span m="66860">dollars.</span> <span m="79330">And</span> <span m="80180">we're</span> <span m="80380">going</span> <span m="80500">to</span> <span m="80580">do</span> <span m="80710">a</span> <span m="80770">simplified</span> <span m="81490">version,</span> <span m="82000">where</span> <span m="82390">in</span> <span m="82580">each</span> <span m="82900">bet,</span> <span m="83970">you</span> <span m="84150">win</span> <span m="84370">$1</span> <span m="85420">or</span> <span m="85590">you</span> <span m="85710">lose</span> <span m="85970">$1.</span> <span m="87080">Now,</span> <span m="87260">these</span> <span m="87490">days,</span> <span m="87750">there are</span> <span m="87810">not</span> <span m="88040">many</span> <span m="88260">bets</span> <span m="88570">in a</span> <span m="88650">casino</span> <span m="89100">for</span> <span m="89230">$1.</span> <span m="89680">It's</span> <span m="89790">more</span> <span m="89940">like</span> <span m="90160">$10.</span> </p>
<p><span m="91150">But</span> <span m="91330">just</span> <span m="91490">to</span> <span m="91570">make</span> <span m="91740">it</span> <span m="91860">simple</span> <span m="92670">for</span> <span m="92820">counting,</span> <span m="94250">we're</span> <span m="94420">going</span> <span m="94520">to</span> <span m="94560">assume</span> <span m="94950">that</span> <span m="95070">each</span> <span m="95370">bet</span> <span m="97980">you</span> <span m="98170">win</span> <span m="99280">$1</span> <span m="100960">with</span> <span m="101190">probability</span> <span m="101790">p,</span> <span m="104800">and</span> <span m="105000">you</span> <span m="105160">lose</span> <span m="106720">$1</span> <span m="108550">with</span> <span m="108930">probability</span> <span m="110930">1</span> <span m="111170">minus</span> <span m="111560">p.</span> <span m="113900">And</span> <span m="114930">in</span> <span m="115070">this</span> <span m="115310">version,</span> <span m="116250">we're</span> <span m="116470">going</span> <span m="116600">to</span> <span m="116640">assume</span> <span m="117120">you</span> <span m="117320">keep</span> <span m="117690">playing</span> <span m="119530">until</span> <span m="119840">one</span> <span m="120040">of</span> <span m="120140">two</span> <span m="120320">things</span> <span m="120610">happens--</span> <span m="121850">you</span> <span m="122000">get</span> <span m="122190">ahead</span> <span m="122690">by</span> <span m="122940">m</span> <span m="123270">dollars,</span> <span m="124380">or</span> <span m="124590">you</span> <span m="124750">lose</span> <span m="125420">all</span> <span m="125680">the</span> <span m="125750">money you</span> <span m="126100">came</span> <span m="126420">with--</span> <span m="126820">all</span> <span m="127140">n</span> <span m="127589">dollars.</span> </p>
<p><span m="130530">So</span> <span m="130720">you</span> <span m="130840">play</span> <span m="131220">until</span> <span m="132950">you</span> <span m="134990">win</span> <span m="136910">m</span> <span m="137170">more--</span> <span m="138040">net</span> <span m="138310">m</span> <span m="138530">plus--</span> <span m="139850">or</span> <span m="140280">you</span> <span m="140470">lose</span> <span m="142230">n.</span> <span m="143860">And</span> <span m="144020">that's</span> <span m="144250">where</span> <span m="144370">you</span> <span m="144470">go</span> <span m="144640">broke.</span> <span m="145030">You</span> <span m="145170">run</span> <span m="145370">out</span> <span m="145490">of</span> <span m="145570">money.</span> <span m="148760">And</span> <span m="148910">we're</span> <span m="149050">going</span> <span m="149130">to</span> <span m="149170">assume</span> <span m="149450">you</span> <span m="149530">don't</span> <span m="149730">borrow</span> <span m="150120">anything</span> <span m="150460">from</span> <span m="150590">the</span> <span m="150650">house.</span> </p>
<p><span m="153750">All</span> <span m="153830">right,</span> <span m="154040">and we're</span> <span m="154200">going</span> <span m="154300">to</span> <span m="154350">look</span> <span m="154620">at</span> <span m="154690">the</span> <span m="154780">probability</span> <span m="156610">that</span> <span m="156760">you</span> <span m="156880">come</span> <span m="157110">out</span> <span m="157290">a</span> <span m="157350">winner</span> <span m="157960">versus</span> <span m="158580">going</span> <span m="158840">home</span> <span m="159070">broke--</span> <span m="159870">that</span> <span m="160000">you</span> <span m="160440">made</span> <span m="160740">m</span> <span m="160970">dollars.</span> <span m="162690">Now,</span> <span m="162940">the</span> <span m="163070">game</span> <span m="163380">we're</span> <span m="163490">going</span> <span m="163590">to</span> <span m="163650">analyze</span> <span m="165330">is</span> <span m="165570">roulette,</span> <span m="166510">but</span> <span m="166790">the</span> <span m="166890">technique</span> <span m="167390">works</span> <span m="167650">for</span> <span m="167790">any</span> <span m="167970">of</span> <span m="168040">them.</span> <span m="174160">How</span> <span m="174280">many</span> <span m="174420">people have</span> <span m="174820">played</span> <span m="175320">roulette</span> <span m="175710">before</span> <span m="176045">in</span> <span m="176380">some</span> <span m="176670">form</span> <span m="176900">or</span> <span m="176920">another?</span> </p>
<p><span m="177850">OK,</span> <span m="178670">so</span> <span m="178830">this</span> <span m="179030">is</span> <span m="179170">a</span> <span m="179210">game</span> <span m="179670">where</span> <span m="180010">there's</span> <span m="180200">the</span> <span m="180290">ball</span> <span m="180660">that</span> <span m="180770">goes</span> <span m="181000">around</span> <span m="181330">the</span> <span m="181400">dish,</span> <span m="181770">and</span> <span m="181830">you</span> <span m="181920">spin</span> <span m="182310">the</span> <span m="182400">wheel.</span> <span m="183310">And</span> <span m="183580">there's</span> <span m="183870">36</span> <span m="184510">numbers</span> <span m="186050">from</span> <span m="186230">1</span> <span m="186430">to</span> <span m="186500">36.</span> <span m="187170">Half</span> <span m="187490">of</span> <span m="187650">them are</span> <span m="187770">red,</span> <span m="188160">half</span> <span m="188450">are</span> <span m="188500">black.</span> <span m="189260">And</span> <span m="189450">then</span> <span m="189540">there's</span> <span m="189760">the</span> <span m="189850">zero</span> <span m="190310">and</span> <span m="190370">the</span> <span m="190470">double</span> <span m="190780">zero</span> <span m="191070">that</span> <span m="191200">are</span> <span m="191270">green.</span> </p>
<p><span m="192810">And</span> <span m="193290">we're</span> <span m="193430">going</span> <span m="193540">to</span> <span m="193580">look</span> <span m="193720">at</span> <span m="193790">the</span> <span m="193870">version</span> <span m="194310">where</span> <span m="194470">you</span> <span m="194620">just</span> <span m="194930">bet</span> <span m="195260">on</span> <span m="195480">red</span> <span m="195800">or</span> <span m="195890">black.</span> <span m="197000">And</span> <span m="197300">you</span> <span m="197420">win</span> <span m="197810">if</span> <span m="197900">the</span> <span m="197980">ball</span> <span m="198360">lands</span> <span m="198710">on</span> <span m="198800">a</span> <span m="198850">slot</span> <span m="199190">that's</span> <span m="199350">red.</span> <span m="199710">And</span> <span m="199810">there's</span> <span m="200110">18</span> <span m="200530">of</span> <span m="200590">those.</span> <span m="201330">And</span> <span m="201490">you</span> <span m="201630">lose</span> <span m="202150">otherwise.</span> </p>
<p><span m="203690">So</span> <span m="204080">in</span> <span m="204150">this</span> <span m="204310">case,</span> <span m="204620">the</span> <span m="204720">probability</span> <span m="205180">of</span> <span m="205270">winning,</span> <span m="205940">p,</span> <span m="208530">is</span> <span m="208820">there's</span> <span m="209090">18</span> <span m="209630">chances</span> <span m="210090">to</span> <span m="210220">win.</span> <span m="211520">And</span> <span m="211770">it's</span> <span m="212130">not</span> <span m="212580">36</span> <span m="213400">total.</span> <span m="213760">It's</span> <span m="213940">38</span> <span m="214590">total</span> <span m="215400">because</span> <span m="215750">of</span> <span m="215840">the</span> <span m="215920">zero</span> <span m="216360">and</span> <span m="216410">the</span> <span m="216500">double</span> <span m="216800">zero.</span> <span m="218200">All</span> <span m="218290">right</span> <span m="218570">so</span> <span m="218810">this</span> <span m="219050">is</span> <span m="220340">9/19</span> <span m="221680">chance</span> <span m="222190">of</span> <span m="222290">winning</span> <span m="222680">and</span> <span m="222770">a</span> <span m="222860">10/19</span> <span m="223920">chance</span> <span m="224280">of</span> <span m="224390">losing.</span> </p>
<p><span m="225960">And</span> <span m="226220">so</span> <span m="226320">this</span> <span m="226500">is</span> <span m="226600">a</span> <span m="226650">game</span> <span m="227090">that</span> <span m="227510">has</span> <span m="227850">a</span> <span m="228620">chance</span> <span m="228960">of</span> <span m="229060">winning of</span> <span m="229320">about</span> <span m="229650">47%,</span> <span m="231400">so</span> <span m="231570">it's</span> <span m="231680">almost</span> <span m="232310">a</span> <span m="232360">fair</span> <span m="232700">game.</span> <span m="233370">It's</span> <span m="233530">not</span> <span m="233810">50-50.</span> <span m="235070">And</span> <span m="235200">that's</span> <span m="235370">because</span> <span m="235660">the</span> <span m="235750">casino's</span> <span m="236210">got</span> <span m="236350">to</span> <span m="236410">make</span> <span m="236640">some</span> <span m="236780">money.</span> </p>
<p><span m="237160">I</span> <span m="237190">mean,</span> <span m="237800">they</span> <span m="238310">have</span> <span m="238510">the</span> <span m="238580">big</span> <span m="238800">facility.</span> <span m="239400">They're</span> <span m="239520">giving</span> <span m="239770">you</span> <span m="239880">free</span> <span m="240140">drinks,</span> <span m="240680">and</span> <span m="240770">all</span> <span m="240990">the</span> <span m="241090">rest.</span> <span m="242010">So</span> <span m="242130">they</span> <span m="242230">got</span> <span m="242400">to</span> <span m="242450">make</span> <span m="242620">money</span> <span m="242870">somehow.</span> </p>
<p><span m="243880">And</span> <span m="244070">they</span> <span m="244170">make</span> <span m="244420">money</span> <span m="245020">on</span> <span m="245220">this</span> <span m="245480">bet</span> <span m="245790">because</span> <span m="247120">they're</span> <span m="247440">going</span> <span m="247550">to</span> <span m="248180">make</span> <span m="249840">$0.03</span> <span m="251590">on</span> <span m="251790">the</span> <span m="251890">dollar</span> <span m="252270">here.</span> <span m="252430">You're</span> <span m="252550">going</span> <span m="252660">to</span> <span m="253500">wager.</span> <span m="253780">And</span> <span m="254060">then</span> <span m="254220">you're</span> <span m="254320">going</span> <span m="254420">to</span> <span m="254470">come</span> <span m="254660">back</span> <span m="255330">with</span> <span m="255580">47%.</span> <span m="258230">And</span> <span m="258410">people</span> <span m="258730">generally</span> <span m="259120">are</span> <span m="259180">fine</span> <span m="259459">with</span> <span m="259579">that.</span> <span m="260339">They</span> <span m="260390">don't</span> <span m="260600">expect</span> <span m="261220">to</span> <span m="261290">have</span> <span m="261519">the</span> <span m="261670">odds</span> <span m="261930">in</span> <span m="261990">their</span> <span m="262110">favor</span> <span m="262900">when</span> <span m="263020">you're</span> <span m="263150">gambling</span> <span m="263440">in</span> <span m="263730">a</span> <span m="263970">casino.</span> </p>
<p><span m="266010">Now,</span> <span m="266360">in</span> <span m="266500">an</span> <span m="266590">effort</span> <span m="266950">to</span> <span m="267280">sort</span> <span m="267580">of</span> <span m="268530">come</span> <span m="268780">home</span> <span m="268990">a</span> <span m="269060">winner,</span> <span m="270180">the</span> <span m="270290">way</span> <span m="270440">people</span> <span m="270780">do</span> <span m="270960">that--</span> <span m="271200">knowing</span> <span m="271640">that</span> <span m="272000">the</span> <span m="272200">odds</span> <span m="272470">are</span> <span m="272780">a</span> <span m="272900">little</span> <span m="273220">against</span> <span m="273630">them--</span> <span m="274740">is</span> <span m="275420">they</span> <span m="275540">might</span> <span m="275980">put</span> <span m="276220">more</span> <span m="276860">money</span> <span m="277210">in</span> <span m="277330">their</span> <span m="277460">pocket</span> <span m="277850">coming</span> <span m="278210">in</span> <span m="278580">than</span> <span m="278720">they</span> <span m="278880">expect</span> <span m="279570">to</span> <span m="279690">win.</span> <span m="280050">So</span> <span m="280940">often,</span> <span m="281250">you'll</span> <span m="281390">see</span> <span m="281650">people</span> <span m="282140">come</span> <span m="282400">into</span> <span m="282770">the</span> <span m="283250">casino</span> <span m="283760">with</span> <span m="283880">the</span> <span m="283970">goal</span> <span m="284650">of</span> <span m="284730">winning</span> <span m="285150">100,</span> <span m="287370">but</span> <span m="287540">they</span> <span m="287660">start</span> <span m="288100">with</span> <span m="288390">1,000</span> <span m="289010">in</span> <span m="289090">their</span> <span m="289220">pocket.</span> <span m="292730">So</span> <span m="292880">they're</span> <span m="293010">willing</span> <span m="293360">to</span> <span m="293490">risk</span> <span m="293820">$1,000,</span> <span m="295440">but</span> <span m="295560">they're</span> <span m="295670">going</span> <span m="295790">to</span> <span m="295870">quit</span> <span m="296340">happy</span> <span m="297240">if</span> <span m="297400">they</span> <span m="297520">get</span> <span m="297690">up</span> <span m="297850">100.</span> </p>
<p><span m="299790">OK</span> <span m="301160">so</span> <span m="301610">you</span> <span m="301710">either</span> <span m="301830">go</span> <span m="301960">home</span> <span m="302270">with</span> <span m="302400">$1,100,</span> <span m="303450">or</span> <span m="303560">you're</span> <span m="303660">going</span> <span m="303920">home</span> <span m="304120">with</span> <span m="304280">$0,</span> <span m="304970">in</span> <span m="305050">this</span> <span m="305240">case.</span> <span m="305550">And</span> <span m="305610">you</span> <span m="305690">came</span> <span m="305920">with</span> <span m="306050">$1,000.</span> <span m="307760">And</span> <span m="307930">this</span> <span m="308120">means</span> <span m="308540">that</span> <span m="308610">you're--</span> <span m="309290">at</span> <span m="309340">least</span> <span m="309560">the</span> <span m="309580">thinking</span> <span m="309860">goes--</span> <span m="310300">this</span> <span m="310490">means</span> <span m="310850">you're</span> <span m="310970">more</span> <span m="311240">likely</span> <span m="312020">to</span> <span m="312140">go</span> <span m="312280">home</span> <span m="312470">happy.</span> <span m="313890">If</span> <span m="314010">you</span> <span m="314130">quit</span> <span m="314440">when</span> <span m="314550">you</span> <span m="314630">get</span> <span m="314760">up</span> <span m="314880">by</span> <span m="315020">100,</span> <span m="316110">you're</span> <span m="316230">more</span> <span m="316530">likely</span> <span m="316930">to</span> <span m="317020">land</span> <span m="317400">there,</span> <span m="317580">because</span> <span m="317750">it's</span> <span m="317900">almost</span> <span m="318210">a</span> <span m="318260">fair</span> <span m="318490">game,</span> <span m="318830">than</span> <span m="318960">you</span> <span m="319100">are</span> <span m="319240">to</span> <span m="319300">lose</span> <span m="319720">all</span> <span m="319950">1,000.</span> <span m="321650">That's the</span> <span m="321820">thinking</span> <span m="322220">anyway.</span> </p>
<p><span m="323570">In</span> <span m="323650">fact,</span> <span m="324040">my</span> <span m="324190">mother-in-law</span> <span m="325180">plays</span> <span m="325510">roulette,</span> <span m="326220">red</span> <span m="326420">and</span> <span m="326510">black,</span> <span m="327090">and</span> <span m="327290">she</span> <span m="327450">follows</span> <span m="327870">the</span> <span m="327980">strategy.</span> <span m="330040">And</span> <span m="330170">she</span> <span m="330380">claims</span> <span m="330860">that</span> <span m="330980">she</span> <span m="331110">does</span> <span m="331450">this</span> <span m="331980">for</span> <span m="332060">that</span> <span m="332260">reason--</span> <span m="332570">that</span> <span m="332690">she</span> <span m="332850">almost</span> <span m="333330">always</span> <span m="334060">wins.</span> <span m="335510">She</span> <span m="335640">goes</span> <span m="335900">home</span> <span m="336060">happy</span> <span m="336560">almost</span> <span m="336940">always.</span> </p>
<p><span m="337820">And</span> <span m="338000">that's</span> <span m="338320">the</span> <span m="338450">important</span> <span m="338870">thing</span> <span m="339090">here.</span> <span m="340290">And</span> <span m="340370">it</span> <span m="340450">does</span> <span m="340950">reasonable,</span> <span m="341400">because</span> <span m="341670">after</span> <span m="342010">all,</span> <span m="342160">roulette</span> <span m="342500">is</span> <span m="342660">almost</span> <span m="343200">a</span> <span m="343250">fair</span> <span m="344050">game.</span> <span m="345980">So</span> <span m="346040">what</span> <span m="346150">do</span> <span m="346190">you</span> <span m="346320">think?</span> </p>
<p><span m="347470">How</span> <span m="347620">many</span> <span m="347830">people</span> <span m="348180">think</span> <span m="348420">she's</span> <span m="348660">right</span> <span m="349100">that</span> <span m="349230">she</span> <span m="349390">almost</span> <span m="349770">always</span> <span m="350790">wins?</span> <span m="352650">Anybody?</span> <span m="353700">I</span> <span m="353830">have</span> <span m="354040">sort</span> <span m="354340">of</span> <span m="354400">set</span> <span m="354670">it</span> <span m="354740">up.</span> <span m="354900">It's</span> <span m="355010">my</span> <span m="355140">mother-in-law,</span> <span m="355660">after</span> <span m="355990">all,</span> <span m="356775">so</span> <span m="357060">probably</span> <span m="357440">she's</span> <span m="357630">going</span> <span m="357740">to</span> <span m="357800">be</span> <span m="357930">wrong.</span> </p>
<p><span m="361740">Well,</span> <span m="362030">how</span> <span m="362230">many</span> <span m="362480">people</span> <span m="362850">think</span> <span m="363180">it's</span> <span m="363920">better</span> <span m="364200">than</span> <span m="364340">a</span> <span m="364400">50%</span> <span m="365290">chance</span> <span m="366480">you</span> <span m="366660">win</span> <span m="366930">$100</span> <span m="367850">before</span> <span m="368160">you</span> <span m="368300">lose</span> <span m="368860">$1,000?</span> <span m="371610">That's</span> <span m="372520">probably</span> <span m="372920">more--</span> <span m="374520">how</span> <span m="374620">many</span> <span m="374770">people</span> <span m="375030">think</span> <span m="375220">you're</span> <span m="375310">more</span> <span m="375520">likely</span> <span m="375870">to</span> <span m="375940">lose</span> <span m="376140">$1,000</span> <span m="377110">before</span> <span m="377370">you</span> <span m="377470">win</span> <span m="377600">$100?</span> <span m="379350">Wow,</span> <span m="379720">OK,</span> <span m="379990">so</span> <span m="380130">you've</span> <span m="380230">been</span> <span m="380360">to</span> <span m="380420">6.04</span> <span m="380940">too</span> <span m="381120">long</span> <span m="381390">now.</span> </p>
<p><span m="381960">OK,</span> <span m="382760">what</span> <span m="382980">about</span> <span m="383200">this--</span> <span m="383390">how</span> <span m="383540">many</span> <span m="383740">people</span> <span m="384070">think</span> <span m="384310">you're</span> <span m="384420">more</span> <span m="384720">likely</span> <span m="385100">to</span> <span m="385170">lose</span> <span m="385620">$10,000</span> <span m="387960">than</span> <span m="388130">to</span> <span m="388220">win</span> <span m="388430">$100?</span> <span m="390720">All right,</span> <span m="391290">how</span> <span m="391440">many</span> <span m="391610">people</span> <span m="391870">think</span> <span m="392030">you're</span> <span m="392130">more</span> <span m="392370">likely</span> <span m="392680">to</span> <span m="392760">lose</span> <span m="393110">$1</span> <span m="393200">million?</span> <span m="397060">A bunch of</span> <span m="397260">you</span> <span m="397360">still</span> <span m="397620">think</span> <span m="397820">that.</span> </p>
<p><span m="398030">OK,</span> <span m="398710">well,</span> <span m="398870">you're</span> <span m="399060">right.</span> <span m="400370">In</span> <span m="400530">fact,</span> <span m="402000">it</span> <span m="402140">is</span> <span m="402290">almost</span> <span m="402780">certain</span> <span m="404030">you</span> <span m="404190">will</span> <span m="404330">go</span> <span m="404500">broke,</span> <span m="405830">no</span> <span m="405940">matter</span> <span m="406220">how</span> <span m="406630">much</span> <span m="406900">money</span> <span m="407180">you</span> <span m="407280">bring,</span> <span m="408830">before</span> <span m="409140">you</span> <span m="409240">win</span> <span m="409380">$100.</span> <span m="411730">In</span> <span m="411920">fact,</span> <span m="412980">we're</span> <span m="413180">going</span> <span m="413280">to</span> <span m="413360">prove</span> <span m="413760">today</span> <span m="415290">that</span> <span m="415460">the</span> <span m="415550">probability</span> <span m="416890">that</span> <span m="417060">you</span> <span m="417240">win</span> <span m="418540">$100</span> <span m="421080">before</span> <span m="421580">losing</span> <span m="422080">$100</span> <span m="422670">million</span> <span m="423190">if</span> <span m="423660">you</span> <span m="423750">stayed</span> <span m="424020">long</span> <span m="424340">enough--</span> <span m="424670">that</span> <span m="424870">takes</span> <span m="425090">a</span> <span m="425160">while--</span> <span m="426800">the</span> <span m="426910">chance</span> <span m="427270">you</span> <span m="427490">go</span> <span m="427620">home</span> <span m="427780">a</span> <span m="427850">winner</span> <span m="429160">is</span> <span m="429330">less</span> <span m="429590">than</span> <span m="429740">1</span> <span m="429930">in</span> <span m="430020">37,648.</span> <span m="434470">You</span> <span m="434640">have</span> <span m="434800">no</span> <span m="435020">chance</span> <span m="435400">to</span> <span m="435490">go</span> <span m="435640">home</span> <span m="435860">happy.</span> </p>
<p><span m="438480">So</span> <span m="438650">my</span> <span m="438790">mother-in-law's</span> <span m="439480">telling</span> <span m="439750">me</span> <span m="439830">the</span> <span m="439940">story</span> <span m="440200">about</span> <span m="440370">how</span> <span m="440490">she</span> <span m="440700">always</span> <span m="440950">goes</span> <span m="441160">home</span> <span m="441340">happy.</span> <span m="441740">And</span> <span m="441830">I'm</span> <span m="441900">saying,</span> <span m="442730">no,</span> <span m="442860">no,</span> <span m="443060">wait</span> <span m="443270">a</span> <span m="443300">minute,</span> <span m="444330">you</span> <span m="444500">can't.</span> <span m="444960">You</span> <span m="445160">never</span> <span m="445450">went</span> <span m="445660">home</span> <span m="445890">happy.</span> <span m="446230">Let's</span> <span m="446440">be</span> <span m="446620">honest.</span> <span m="448040">It</span> <span m="448130">can't</span> <span m="448480">be.</span> </p>
<p><span m="449250">She goes,</span> <span m="449470">no,</span> <span m="449700">no,</span> <span m="449830">no,</span> <span m="450080">it's</span> <span m="450170">true.</span> <span m="450390">I go,</span> <span m="450550">no,</span> <span m="451040">look,</span> <span m="451770">there's</span> <span m="451910">a</span> <span m="451960">mathematical</span> <span m="452510">proof.</span> <span m="452940">I</span> <span m="453020">have</span> <span m="453170">a</span> <span m="453240">proof.</span> <span m="453530">I</span> <span m="453600">can</span> <span m="453760">show</span> <span m="453920">you</span> <span m="454060">my</span> <span m="454190">proof--</span> <span m="455010">very</span> <span m="455450">unlikely</span> <span m="455705">you</span> <span m="455960">go</span> <span m="456130">home a</span> <span m="456370">winner.</span> </p>
<p><span m="458140">So</span> <span m="458900">somehow,</span> <span m="459210">she's</span> <span m="459410">not</span> <span m="459580">very</span> <span m="459760">impressed</span> <span m="460150">with the</span> <span m="460260">mathematical</span> <span m="460790">proof.</span> <span m="462300">And</span> <span m="462390">she</span> <span m="462580">keeps</span> <span m="462810">insisting.</span> <span m="463320">And</span> <span m="463410">I</span> <span m="463470">keep</span> <span m="463690">trying</span> <span m="463940">to</span> <span m="463990">show</span> <span m="464220">her</span> <span m="464320">the</span> <span m="464460">proof.</span> <span m="465400">And</span> <span m="465910">anyway,</span> <span m="466300">I</span> <span m="466400">hope</span> <span m="466620">I'll</span> <span m="466660">have</span> <span m="466850">more</span> <span m="467040">luck</span> <span m="467220">with</span> <span m="467350">you</span> <span m="467470">guys</span> <span m="467740">today</span> <span m="468170">in</span> <span m="468270">showing</span> <span m="468570">you</span> <span m="468650">the</span> <span m="468770">proof</span> <span m="469950">that</span> <span m="470310">the</span> <span m="470390">chance</span> <span m="470700">you</span> <span m="470780">go</span> <span m="470940">home</span> <span m="471150">happy</span> <span m="471520">here</span> <span m="472230">is</span> <span m="472490">very,</span> <span m="473410">very</span> <span m="473620">small.</span> <span m="474600">Now,</span> <span m="474660">in</span> <span m="474720">the</span> <span m="474830">end,</span> <span m="474970">I</span> <span m="475020">didn't</span> <span m="475270">convince</span> <span m="475640">her,</span> <span m="476240">but</span> <span m="476710">we'll</span> <span m="477000">see</span> <span m="477290">how</span> <span m="477440">we</span> <span m="477550">do</span> <span m="477690">here</span> <span m="477870">today.</span> </p>
<p><span m="479680">Now,</span> <span m="480170">in</span> <span m="480290">order</span> <span m="480540">to</span> <span m="480650">see</span> <span m="481000">why</span> <span m="481200">this</span> <span m="481370">probability</span> <span m="481900">is</span> <span m="482010">so</span> <span m="482200">stunningly</span> <span m="482940">small--</span> <span m="484290">you</span> <span m="484460">would</span> <span m="484580">just</span> <span m="484770">never</span> <span m="485040">guess</span> <span m="485360">it's</span> <span m="485760">that</span> <span m="485950">low--</span> <span m="486690">we've</span> <span m="486910">got</span> <span m="487080">to</span> <span m="487170">learn</span> <span m="487630">about</span> <span m="487940">random</span> <span m="488320">walks.</span> <span m="489510">And</span> <span m="489690">they</span> <span m="489790">come</span> <span m="489990">up</span> <span m="490110">in</span> <span m="490240">all</span> <span m="490600">sorts</span> <span m="491010">of</span> <span m="491110">applications.</span> <span m="492010">In</span> <span m="492140">fact,</span> <span m="493650">page</span> <span m="493990">rank--</span> <span m="494780">that</span> <span m="495070">got</span> <span m="495270">Google</span> <span m="495570">started--</span> <span m="496330">it's</span> <span m="496440">all</span> <span m="496670">based</span> <span m="496990">on</span> <span m="497100">a</span> <span m="497190">random</span> <span m="497650">walk</span> <span m="498480">through</span> <span m="498700">the</span> <span m="498810">Web</span> <span m="499390">or</span> <span m="499540">through</span> <span m="499660">the</span> <span m="499770">links</span> <span m="500830">on</span> <span m="501030">web</span> <span m="501260">pages</span> <span m="501880">on</span> <span m="502040">that</span> <span m="502240">graph.</span> </p>
<p><span m="504600">Now,</span> <span m="505190">for</span> <span m="505280">the</span> <span m="505380">gambling</span> <span m="505790">problem,</span> <span m="506270">we're</span> <span m="506380">going</span> <span m="506460">to</span> <span m="506500">look</span> <span m="506640">at</span> <span m="506710">a</span> <span m="506750">very</span> <span m="507330">special</span> <span m="507850">case--</span> <span m="508180">probably</span> <span m="508320">the</span> <span m="508430">simplest</span> <span m="508890">case</span> <span m="509130">of a</span> <span m="509200">random</span> <span m="509610">walk--</span> <span m="510210">and</span> <span m="510380">that's</span> <span m="510560">a</span> <span m="510630">one-dimensional</span> <span m="511840">random</span> <span m="512210">walk.</span> <span m="513390">In</span> <span m="513510">a</span> <span m="513580">one-dimensional</span> <span m="514360">random</span> <span m="514710">walk,</span> <span m="515049">there's</span> <span m="515240">some</span> <span m="515580">value--</span> <span m="516710">say</span> <span m="516870">the</span> <span m="516990">number</span> <span m="517260">of</span> <span m="517370">dollars</span> <span m="517809">you've</span> <span m="517970">got</span> <span m="519039">in</span> <span m="519140">your</span> <span m="519240">pocket.</span> <span m="520240">And</span> <span m="520429">this</span> <span m="520630">value</span> <span m="521049">can</span> <span m="521190">go</span> <span m="521429">up,</span> <span m="522240">or</span> <span m="522390">go</span> <span m="522549">down,</span> <span m="523309">or</span> <span m="523460">stay</span> <span m="523710">the</span> <span m="523820">same</span> <span m="524490">each</span> <span m="524680">time</span> <span m="524880">you</span> <span m="524970">do</span> <span m="525150">something</span> <span m="525630">like</span> <span m="526000">make</span> <span m="526270">a</span> <span m="526330">bet.</span> <span m="527310">And</span> <span m="527470">each</span> <span m="527660">of</span> <span m="527740">this</span> <span m="528660">happens</span> <span m="529140">with</span> <span m="529300">a</span> <span m="529830">certain</span> <span m="530120">probability.</span> </p>
<p><span m="531760">Now</span> <span m="532010">in</span> <span m="532110">this</span> <span m="532330">case,</span> <span m="533560">you</span> <span m="533700">either</span> <span m="533910">go</span> <span m="534090">up</span> <span m="534370">by</span> <span m="534510">one,</span> <span m="535360">or</span> <span m="535470">you</span> <span m="535550">go</span> <span m="535640">down</span> <span m="535940">by</span> <span m="536050">one,</span> <span m="536650">and</span> <span m="536770">you</span> <span m="536840">can't</span> <span m="537140">stay</span> <span m="537320">the</span> <span m="537450">same.</span> <span m="538010">Every</span> <span m="538280">bet</span> <span m="538560">you</span> <span m="538670">win</span> <span m="538870">$1</span> <span m="539230">or</span> <span m="539320">you</span> <span m="539440">lose</span> <span m="539630">$1.</span> <span m="540100">So</span> <span m="540120">it's</span> <span m="540760">really</span> <span m="541040">a</span> <span m="541090">special</span> <span m="541440">case.</span> </p>
<p><span m="542490">And</span> <span m="542650">we</span> <span m="542770">can</span> <span m="542940">diagram</span> <span m="543590">it</span> <span m="544760">as</span> <span m="545020">follows.</span> <span m="545700">We</span> <span m="545720">can</span> <span m="545840">put</span> <span m="546620">time,</span> <span m="547860">or</span> <span m="547980">the</span> <span m="548090">number</span> <span m="548510">of bets,</span> <span m="549090">on</span> <span m="549220">this</span> <span m="549430">axis.</span> <span m="552900">And</span> <span m="553310">we</span> <span m="553420">can</span> <span m="553530">put</span> <span m="554260">the</span> <span m="554370">number</span> <span m="554600">of</span> <span m="554670">dollars</span> <span m="555680">on</span> <span m="555830">this</span> <span m="556050">axis.</span> </p>
<p><span m="560120">Now</span> <span m="560370">in</span> <span m="560460">this</span> <span m="560660">case,</span> <span m="562230">we</span> <span m="562330">start</span> <span m="562700">with</span> <span m="562830">n</span> <span m="563010">dollars.</span> <span m="565010">And</span> <span m="565540">we</span> <span m="565660">might</span> <span m="566580">win</span> <span m="566830">the</span> <span m="566910">first</span> <span m="567270">bet,</span> <span m="567570">so</span> <span m="567650">we</span> <span m="567740">go</span> <span m="567850">to</span> <span m="567910">n</span> <span m="568060">plus</span> <span m="568370">1.</span> <span m="568990">We</span> <span m="569060">might</span> <span m="569320">lose</span> <span m="569610">a</span> <span m="569660">bet,</span> <span m="571040">might</span> <span m="571270">lose</span> <span m="571460">again,</span> <span m="573020">could</span> <span m="573170">win</span> <span m="573370">the</span> <span m="573450">next</span> <span m="573750">one,</span> <span m="575240">lose,</span> <span m="576134">win,</span> <span m="577880">lose,</span> <span m="578820">lose.</span> <span m="579770">So</span> <span m="579910">this</span> <span m="580080">corresponds</span> <span m="580660">to</span> <span m="580750">a</span> <span m="580820">string--</span> <span m="581620">win,</span> <span m="582076">lose,</span> <span m="583650">lose,</span> <span m="584820">lose,</span> <span m="585210">lose,</span> <span m="586485">win,</span> <span m="586910">lose,</span> <span m="588065">win,</span> <span m="588450">lose,</span> <span m="589416">lose,</span> <span m="591120">lose.</span> </p>
<p><span m="593290">And</span> <span m="593560">when we</span> <span m="593680">win,</span> <span m="594070">we</span> <span m="594190">go</span> <span m="594410">up</span> <span m="594570">$1.</span> <span m="597180">When</span> <span m="597690">we</span> <span m="597800">lose,</span> <span m="598260">we</span> <span m="598380">go</span> <span m="598530">down</span> <span m="598830">$1.</span> <span m="601790">And</span> <span m="601930">it's</span> <span m="602030">called</span> <span m="602250">one-dimensional,</span> <span m="602980">because</span> <span m="603140">there's</span> <span m="603260">just</span> <span m="603500">one</span> <span m="604150">thing</span> <span m="604400">that's</span> <span m="604580">changing.</span> <span m="605050">You're</span> <span m="605150">going</span> <span m="605380">up</span> <span m="605520">and</span> <span m="605640">down</span> <span m="606280">there.</span> </p>
<p><span m="607510">Now,</span> <span m="607620">the</span> <span m="607730">probability</span> <span m="608290">of</span> <span m="608380">going</span> <span m="608730">up</span> <span m="609690">is</span> <span m="609880">p.</span> <span m="611420">And</span> <span m="611800">that's</span> <span m="612070">no</span> <span m="612190">matter</span> <span m="612470">what</span> <span m="612830">happened</span> <span m="613290">before.</span> <span m="613990">It's</span> <span m="614130">a</span> <span m="614190">memoryless</span> <span m="615530">independent</span> <span m="616510">system.</span> </p>
<p><span m="617380">The</span> <span m="617480">probability</span> <span m="618010">you</span> <span m="618150">win</span> <span m="618480">your</span> <span m="618690">i-th</span> <span m="618970">bet</span> <span m="619920">has</span> <span m="620160">nothing</span> <span m="620590">to</span> <span m="620670">do--</span> <span m="620950">is</span> <span m="621040">totally</span> <span m="621400">independent,</span> <span m="621960">mutually</span> <span m="622370">independent--</span> <span m="622820">of</span> <span m="623010">all</span> <span m="623380">the</span> <span m="623480">other</span> <span m="623630">bets</span> <span m="623940">that</span> <span m="624080">took</span> <span m="624270">place</span> <span m="624970">before.</span> <span m="626900">So</span> <span m="627080">let's</span> <span m="628000">write</span> <span m="628230">that</span> <span m="628450">down.</span> <span m="632410">So</span> <span m="632560">the</span> <span m="632650">probability</span> <span m="634040">of</span> <span m="634200">an</span> <span m="634350">up</span> <span m="634620">move</span> <span m="638210">is</span> <span m="638430">p.</span> <span m="640040">The</span> <span m="640130">probability</span> <span m="640920">of</span> <span m="641010">a</span> <span m="641070">down</span> <span m="641470">move</span> <span m="644190">is</span> <span m="644390">1</span> <span m="644640">minus</span> <span m="645030">p.</span> <span m="646210">And</span> <span m="646420">these</span> <span m="646670">are</span> <span m="646790">mutually</span> <span m="647280">independent</span> <span m="651124">of</span> <span m="651600">past</span> <span m="652220">moves.</span> </p>
<p><span m="656840">Now,</span> <span m="657040">when you</span> <span m="657210">have</span> <span m="657500">a</span> <span m="657610">random</span> <span m="657990">walk</span> <span m="658940">where</span> <span m="659080">the</span> <span m="659150">moves</span> <span m="659700">are</span> <span m="659870">mutually</span> <span m="660320">independent,</span> <span m="661060">it</span> <span m="661160">has</span> <span m="661360">a</span> <span m="661430">special</span> <span m="661850">name.</span> <span m="662740">It's</span> <span m="663010">called</span> <span m="663280">a</span> <span m="663340">martingale.</span> <span m="670120">All</span> <span m="670380">random</span> <span m="670670">walks</span> <span m="671260">don't</span> <span m="671470">have</span> <span m="671820">to</span> <span m="671910">have</span> <span m="672070">mutually</span> <span m="672380">independent</span> <span m="672810">steps.</span> </p>
<p><span m="673850">Say</span> <span m="674010">you're</span> <span m="674190">looking</span> <span m="674420">about</span> <span m="674630">winning</span> <span m="674910">and</span> <span m="674980">losing</span> <span m="675330">a</span> <span m="675370">baseball</span> <span m="675850">game</span> <span m="676120">in a</span> <span m="676210">series.</span> <span m="677460">We looked</span> <span m="677550">at</span> <span m="677730">a</span> <span m="677830">scenario</span> <span m="678350">where,</span> <span m="678520">if</span> <span m="678590">you</span> <span m="678690">lost</span> <span m="679030">yesterday,</span> <span m="679630">you're</span> <span m="679740">feeling</span> <span m="680030">lousy,</span> <span m="680470">more</span> <span m="680650">likely to</span> <span m="680950">lose</span> <span m="681210">today.</span> <span m="682310">Not</span> <span m="682470">true</span> <span m="682820">in</span> <span m="683170">the</span> <span m="683280">gambling</span> <span m="683690">case</span> <span m="683980">here.</span> <span m="684140">It's</span> <span m="684270">mutually</span> <span m="684730">independent.</span> <span m="685680">And</span> <span m="685780">that's</span> <span m="685980">the</span> <span m="686080">only</span> <span m="686300">case</span> <span m="686560">we're</span> <span m="686670">going</span> <span m="686760">to</span> <span m="686830">study</span> <span m="687120">for</span> <span m="687260">random</span> <span m="687560">walks.</span> </p>
<p><span m="689290">Now,</span> <span m="691580">if</span> <span m="692050">p</span> <span m="692250">is</span> <span m="692400">not</span> <span m="692650">1/2,</span> <span m="694810">the</span> <span m="694960">random</span> <span m="695340">walk</span> <span m="695740">is</span> <span m="695840">said</span> <span m="696050">to</span> <span m="696110">be</span> <span m="696230">biased.</span> <span m="703720">And</span> <span m="703830">that's</span> <span m="704090">what</span> <span m="704170">happens</span> <span m="704620">in</span> <span m="704680">the</span> <span m="704770">casino.</span> <span m="705430">It's</span> <span m="705700">biased</span> <span m="706420">in</span> <span m="706520">favor</span> <span m="706890">of</span> <span m="706960">the</span> <span m="707020">house.</span> <span m="709040">If</span> <span m="709210">p</span> <span m="709540">equals</span> <span m="709930">1/2,</span> <span m="711740">then</span> <span m="712280">the</span> <span m="712370">random</span> <span m="712680">walk</span> <span m="713070">is</span> <span m="713250">unbiased.</span> </p>
<p><span m="720440">Now,</span> <span m="720680">in</span> <span m="720770">this</span> <span m="720940">particular</span> <span m="721450">case</span> <span m="721770">that</span> <span m="721880">we're</span> <span m="721980">looking</span> <span m="722340">at,</span> <span m="723360">we</span> <span m="723610">have</span> <span m="724260">boundaries</span> <span m="725310">on</span> <span m="725500">the</span> <span m="725590">random</span> <span m="725950">walk.</span> <span m="727070">There's</span> <span m="727260">a</span> <span m="727320">boundary</span> <span m="728010">at</span> <span m="728600">0,</span> <span m="729710">because</span> <span m="729910">you</span> <span m="730000">go</span> <span m="730220">home</span> <span m="730460">broke</span> <span m="731280">if</span> <span m="731420">you</span> <span m="731550">lost</span> <span m="731850">everything.</span> <span m="732220">If</span> <span m="732310">the</span> <span m="732390">random</span> <span m="732690">walk</span> <span m="732980">ever</span> <span m="733180">hit</span> <span m="733400">$0,</span> <span m="734520">you're</span> <span m="734640">done.</span> </p>
<p><span m="735770">And</span> <span m="735950">we're</span> <span m="736100">also</span> <span m="737090">going</span> <span m="737210">to</span> <span m="737290">put</span> <span m="737520">a</span> <span m="737570">boundary</span> <span m="741250">at</span> <span m="742150">n</span> <span m="742430">plus</span> <span m="742780">m.</span> <span m="746220">So</span> <span m="746340">I'm</span> <span m="746420">going</span> <span m="746510">to</span> <span m="746550">have</span> <span m="746670">a</span> <span m="746730">boundary</span> <span m="748920">here.</span> <span m="750690">So</span> <span m="750840">that</span> <span m="751710">if</span> <span m="751870">I</span> <span m="751960">win</span> <span m="752420">m</span> <span m="752670">dollars</span> <span m="753270">here,</span> <span m="754410">I</span> <span m="754510">stop</span> <span m="756170">and</span> <span m="756290">I</span> <span m="756350">go</span> <span m="756540">home</span> <span m="756710">happy.</span> <span m="757180">If the</span> <span m="757280">random</span> <span m="757550">walk</span> <span m="757840">ever</span> <span m="758070">goes</span> <span m="758340">here,</span> <span m="759090">then</span> <span m="759250">I</span> <span m="759320">stop.</span> </p>
<p><span m="761790">Those</span> <span m="761990">are</span> <span m="762050">called</span> <span m="762450">boundary</span> <span m="762940">conditions</span> <span m="763640">for</span> <span m="763790">the</span> <span m="763890">walk.</span> <span m="765970">And</span> <span m="766130">what</span> <span m="766210">we</span> <span m="766310">want</span> <span m="766550">to</span> <span m="766630">do</span> <span m="767310">is</span> <span m="767490">analyze</span> <span m="768110">the</span> <span m="768210">probability</span> <span m="769800">that</span> <span m="769960">we</span> <span m="770110">hit</span> <span m="770440">that</span> <span m="770670">top</span> <span m="770940">boundary</span> <span m="771990">before</span> <span m="772400">we</span> <span m="772940">hit</span> <span m="773130">the</span> <span m="773210">bottom</span> <span m="773500">boundary.</span> <span m="775630">So</span> <span m="775850">we're</span> <span m="775950">going</span> <span m="776040">to</span> <span m="776190">define</span> <span m="776520">that</span> <span m="776700">event</span> <span m="777130">to be</span> <span m="777360">W</span> <span m="777740">star.</span> </p>
<p><span m="780580">W</span> <span m="780890">star</span> <span m="781960">is</span> <span m="782140">the</span> <span m="782280">event</span> <span m="784660">that</span> <span m="784800">the</span> <span m="784910">random</span> <span m="785260">walk</span> <span m="785650">hits</span> <span m="789330">T,</span> <span m="790210">which</span> <span m="790410">is</span> <span m="790520">n</span> <span m="790680">plus</span> <span m="790960">m,</span> <span m="793920">before</span> <span m="796840">it</span> <span m="796980">hits</span> <span m="797250">0.</span> <span m="801240">In</span> <span m="801300">other</span> <span m="801460">words,</span> <span m="801750">you</span> <span m="801840">go</span> <span m="802030">home</span> <span m="802220">happy</span> <span m="803440">without</span> <span m="803730">going</span> <span m="803980">broke.</span> <span m="806080">Let's</span> <span m="806350">also</span> <span m="806640">define</span> <span m="807140">D</span> <span m="808380">to</span> <span m="808820">be</span> <span m="808920">the</span> <span m="809010">number</span> <span m="809330">of</span> <span m="809430">dollars</span> <span m="810200">at</span> <span m="810290">the</span> <span m="810390">start.</span> <span m="812020">And</span> <span m="812120">this</span> <span m="812310">is</span> <span m="812540">just</span> <span m="813550">going</span> <span m="813680">to</span> <span m="813730">be</span> <span m="813940">n</span> <span m="814220">in</span> <span m="814340">our</span> <span m="814470">case.</span> </p>
<p><span m="820960">We're</span> <span m="821270">interested</span> <span m="821810">in,</span> <span m="824410">call</span> <span m="824510">it</span> <span m="825110">X</span> <span m="825380">sub</span> <span m="825560">n</span> <span m="826090">is</span> <span m="826260">the</span> <span m="826370">probability</span> <span m="828300">that</span> <span m="828530">we</span> <span m="828810">go</span> <span m="829080">home</span> <span m="829310">happy</span> <span m="831020">given</span> <span m="831750">we</span> <span m="831870">started</span> <span m="832720">with</span> <span m="832920">n</span> <span m="833335">dollars.</span> <span m="836330">And</span> <span m="836410">that's</span> <span m="836610">a</span> <span m="836890">function</span> <span m="837350">of</span> <span m="837450">n.</span> <span m="837740">So</span> <span m="837900">we'll</span> <span m="838310">make</span> <span m="838510">a</span> <span m="838560">variable</span> <span m="839010">called</span> <span m="839300">X n.</span> </p>
<p><span m="840650">And</span> <span m="840730">we</span> <span m="840820">want</span> <span m="841030">to</span> <span m="841070">know</span> <span m="841150">what</span> <span m="841260">that</span> <span m="841410">probability</span> <span m="842090">is.</span> <span m="842710">And</span> <span m="842820">of</span> <span m="842930">course,</span> <span m="843140">the</span> <span m="843210">more</span> <span m="843410">you</span> <span m="843520">come</span> <span m="843820">with,</span> <span m="844020">you'd</span> <span m="844210">think</span> <span m="845590">it's</span> <span m="845760">a</span> <span m="845800">higher</span> <span m="846160">chance</span> <span m="846960">of</span> <span m="847640">winning</span> <span m="848120">the</span> <span m="848190">more</span> <span m="848390">you</span> <span m="848460">have</span> <span m="848660">in</span> <span m="848710">your</span> <span m="848810">pocket,</span> <span m="849150">because</span> <span m="849290">you can</span> <span m="849490">play</span> <span m="849740">for</span> <span m="849870">more.</span> <span m="851250">So</span> <span m="851380">the</span> <span m="851460">goal</span> <span m="851710">is</span> <span m="851770">to</span> <span m="851860">figure</span> <span m="852110">this</span> <span m="852370">out.</span> </p>
<p><span m="854570">Now</span> <span m="854660">to</span> <span m="854790">do</span> <span m="854970">this,</span> <span m="855160">we</span> <span m="855250">could</span> <span m="855380">use</span> <span m="855590">the</span> <span m="855660">tree</span> <span m="855920">method.</span> <span m="858050">But</span> <span m="858260">it</span> <span m="858360">gets</span> <span m="858640">pretty</span> <span m="858930">complicated,</span> <span m="860580">because</span> <span m="861050">the</span> <span m="861150">sample</span> <span m="861580">space</span> <span m="862110">is</span> <span m="862230">the</span> <span m="862330">sample</span> <span m="862720">space</span> <span m="863030">of</span> <span m="863160">all</span> <span m="863560">one-loss</span> <span m="864150">sequences.</span> <span m="866390">And</span> <span m="866510">how</span> <span m="866710">big</span> <span m="866990">is</span> <span m="867100">that</span> <span m="867310">sample</span> <span m="867670">space?</span> </p>
<p><span m="872106">AUDIENCE: Infinite.</span> </p>
<p><span m="873100">PROFESSOR: Infinite.</span> <span m="875050">I</span> <span m="875200">could</span> <span m="875390">play</span> <span m="875750">forever.</span> <span m="877240">All</span> <span m="877310">right,</span> <span m="877520">now</span> <span m="877690">it</span> <span m="877790">turns</span> <span m="878100">out</span> <span m="878210">the</span> <span m="878300">probability</span> <span m="878800">of</span> <span m="878870">playing</span> <span m="879180">forever</span> <span m="879560">is 0.</span> <span m="880080">And</span> <span m="880170">we</span> <span m="880260">won't</span> <span m="880520">prove</span> <span m="880850">that,</span> <span m="881660">but</span> <span m="881850">there</span> <span m="882060">are</span> <span m="882280">an</span> <span m="882400">infinite</span> <span m="882840">number</span> <span m="883350">of</span> <span m="883460">sample</span> <span m="883830">points.</span> <span m="884150">So</span> <span m="884210">doing</span> <span m="884500">the</span> <span m="884600">tree</span> <span m="884820">method</span> <span m="885200">is a</span> <span m="885250">little</span> <span m="885450">complicated</span> <span m="886050">when</span> <span m="886180">it's</span> <span m="886330">infinite.</span> </p>
<p><span m="889060">So</span> <span m="890050">what</span> <span m="890290">we're</span> <span m="890410">going</span> <span m="890520">to</span> <span m="890610">do</span> <span m="891570">is</span> <span m="891910">use</span> <span m="892130">some</span> <span m="892280">of</span> <span m="892320">the</span> <span m="892400">theorems</span> <span m="892840">we've</span> <span m="892990">proved</span> <span m="893400">over</span> <span m="893500">the</span> <span m="893580">last</span> <span m="893940">few</span> <span m="894120">weeks</span> <span m="894950">and</span> <span m="895230">set</span> <span m="895410">up</span> <span m="895520">a</span> <span m="895600">recurrence</span> <span m="896710">to</span> <span m="896860">find</span> <span m="897340">this</span> <span m="897500">probability.</span> <span m="901100">Now,</span> <span m="902000">I'm</span> <span m="902150">going</span> <span m="902280">to</span> <span m="903160">tell</span> <span m="903370">you</span> <span m="903490">what</span> <span m="903610">the</span> <span m="903690">recurrence</span> <span m="904190">is,</span> <span m="904640">and</span> <span m="904740">then</span> <span m="904980">prove</span> <span m="906410">that</span> <span m="906720">that's</span> <span m="906930">right.</span> <span m="909730">So</span> <span m="909900">I</span> <span m="910020">claim</span> <span m="911960">that</span> <span m="912250">X n</span> <span m="914340">is</span> <span m="914710">0</span> <span m="916150">probability</span> <span m="917580">if</span> <span m="917970">we</span> <span m="918190">start</span> <span m="918470">with</span> <span m="919380">$0.</span> <span m="921320">It's</span> <span m="921500">1</span> <span m="922170">if</span> <span m="922380">we</span> <span m="922470">start</span> <span m="922820">with</span> <span m="922980">T</span> <span m="923170">dollars.</span> <span m="925730">And</span> <span m="926310">it's</span> <span m="926940">p</span> <span m="927530">times</span> <span m="928750">X</span> <span m="929120">n</span> <span m="929440">minus</span> <span m="929920">1</span> <span m="930800">plus</span> <span m="931520">1</span> <span m="931860">minus</span> <span m="932260">p</span> <span m="933460">X</span> <span m="933860">n</span> <span m="934190">plus</span> <span m="934590">1</span> <span m="936580">if</span> <span m="936850">we</span> <span m="936950">start</span> <span m="937560">with</span> <span m="938030">between</span> <span m="941650">$0</span> <span m="942320">and</span> <span m="942500">T</span> <span m="942680">dollars.</span> </p>
<p><span m="947110">All</span> <span m="947180">right,</span> <span m="947410">so</span> <span m="947510">that's</span> <span m="947760">what</span> <span m="947860">I</span> <span m="947930">claim</span> <span m="948730">X</span> <span m="949000">n</span> <span m="949200">is.</span> <span m="949450">And it's,</span> <span m="949730">of</span> <span m="949780">course,</span> <span m="950020">a</span> <span m="950100">recursion</span> <span m="950610">that</span> <span m="950740">I've</span> <span m="950840">set</span> <span m="951040">up</span> <span m="951200">here.</span> <span m="951960">So</span> <span m="952120">let's</span> <span m="952390">see</span> <span m="952580">why</span> <span m="952820">that's</span> <span m="953110">the</span> <span m="953190">case.</span> </p>
<p><span m="958230">OK,</span> <span m="958590">so</span> <span m="959360">let's</span> <span m="959620">check</span> <span m="959960">the</span> <span m="960040">0</span> <span m="960490">case.</span> <span m="960930">X</span> <span m="961230">0</span> <span m="962870">is</span> <span m="963100">the</span> <span m="963200">probability</span> <span m="964560">we</span> <span m="964690">go</span> <span m="964950">home</span> <span m="965150">a</span> <span m="965230">winner</span> <span m="966600">given</span> <span m="967530">we</span> <span m="967650">started</span> <span m="968180">with</span> <span m="968860">$0.</span> <span m="972510">Why</span> <span m="972810">is</span> <span m="972930">that</span> <span m="973120">0?</span> </p>
<p><span m="976987">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="977890">PROFESSOR: What's</span> <span m="978070">that?</span> </p>
<p><span m="979386">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="981180">PROFESSOR: Yeah,</span> <span m="981860">you</span> <span m="982120">started</span> <span m="982740">broke.</span> <span m="983280">You</span> <span m="983370">never</span> <span m="983720">get</span> <span m="983940">off</span> <span m="984150">the</span> <span m="984230">ground,</span> <span m="985110">because</span> <span m="985360">you</span> <span m="985520">quit</span> <span m="985930">as</span> <span m="986030">soon</span> <span m="986200">as</span> <span m="986280">you</span> <span m="986350">have</span> <span m="986470">$0.</span> <span m="987730">So</span> <span m="987840">you</span> <span m="987920">have</span> <span m="988100">no</span> <span m="988500">chance</span> <span m="989780">to</span> <span m="989940">win,</span> <span m="991750">because</span> <span m="991980">you're</span> <span m="992080">broke</span> <span m="992360">to</span> <span m="992470">start.</span> </p>
<p><span m="995020">Let's</span> <span m="995220">check</span> <span m="995390">the</span> <span m="995460">next</span> <span m="995720">case,</span> <span m="996990">X</span> <span m="997650">T--</span> <span m="999700">case</span> <span m="999970">n</span> <span m="1000100">equals</span> <span m="1000400">T--</span> <span m="1001670">is</span> <span m="1001860">the</span> <span m="1001960">probability</span> <span m="1003060">you</span> <span m="1003220">go</span> <span m="1003470">home</span> <span m="1003680">a</span> <span m="1003780">winner</span> <span m="1005280">given</span> <span m="1005570">you</span> <span m="1005750">started</span> <span m="1006280">with</span> <span m="1006430">T</span> <span m="1006670">dollars.</span> <span m="1009310">Why</span> <span m="1009520">is</span> <span m="1009640">that</span> <span m="1009840">1?</span> <span m="1011750">Why is</span> <span m="1011980">that</span> <span m="1012160">certain,</span> <span m="1013570">sort</span> <span m="1013750">of</span> <span m="1013910">from the</span> <span m="1014040">definition?</span> </p>
<p><span m="1015147">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1016580">PROFESSOR: You</span> <span m="1016750">already</span> <span m="1017060">have</span> <span m="1017380">your</span> <span m="1017480">money.</span> <span m="1018230">You</span> <span m="1018410">already</span> <span m="1018690">hit</span> <span m="1018890">the</span> <span m="1019000">top</span> <span m="1019250">boundary,</span> <span m="1019560">because</span> <span m="1019730">you</span> <span m="1019850">started</span> <span m="1020410">there.</span> <span m="1020630">Remember,</span> <span m="1020840">you</span> <span m="1020990">quit</span> <span m="1021780">and</span> <span m="1021910">you're</span> <span m="1022040">happy.</span> <span m="1022440">Go</span> <span m="1022630">home</span> <span m="1022830">happy</span> <span m="1023140">if</span> <span m="1023220">you</span> <span m="1023340">hit</span> <span m="1023590">T</span> <span m="1023750">dollars.</span> <span m="1025900">All</span> <span m="1025980">right,</span> <span m="1026270">so</span> <span m="1027500">you're</span> <span m="1027660">guaranteed</span> <span m="1028240">to</span> <span m="1028290">go</span> <span m="1028430">home</span> <span m="1028619">happy,</span> <span m="1028810">because</span> <span m="1029030">you</span> <span m="1029109">never</span> <span m="1029310">make</span> <span m="1029500">any</span> <span m="1029680">bets.</span> <span m="1030030">You</span> <span m="1030140">started</span> <span m="1030450">with</span> <span m="1030550">all</span> <span m="1030690">the</span> <span m="1030760">money</span> <span m="1031040">you</span> <span m="1031119">needed</span> <span m="1031369">to</span> <span m="1031420">go</span> <span m="1031599">home</span> <span m="1031790">happy.</span> </p>
<p><span m="1033530">Then</span> <span m="1033670">we</span> <span m="1033770">have</span> <span m="1033970">the</span> <span m="1034099">interesting</span> <span m="1034510">case,</span> <span m="1038490">where</span> <span m="1038700">you</span> <span m="1038829">start</span> <span m="1039130">with</span> <span m="1039250">between</span> <span m="1039730">0</span> <span m="1040099">and</span> <span m="1040210">T</span> <span m="1040400">dollars.</span> <span m="1040849">And</span> <span m="1040940">now</span> <span m="1041089">you're</span> <span m="1041200">going</span> <span m="1041290">to</span> <span m="1041329">make</span> <span m="1041569">some</span> <span m="1041740">bets.</span> <span m="1043990">And</span> <span m="1044119">then</span> <span m="1044460">X</span> <span m="1044725">n</span> <span m="1044990">is</span> <span m="1045140">the</span> <span m="1045230">probability--</span> <span m="1046109">just</span> <span m="1046310">the</span> <span m="1046390">definition--</span> <span m="1047670">of</span> <span m="1047800">going</span> <span m="1048079">home</span> <span m="1048319">happy--</span> <span m="1049710">i.e.</span> <span m="1050180">winning</span> <span m="1050590">and</span> <span m="1050820">having</span> <span m="1051070">T</span> <span m="1051405">dollars,</span> <span m="1052370">if</span> <span m="1052510">you</span> <span m="1052610">start</span> <span m="1052910">with</span> <span m="1053020">n.</span> </p>
<p><span m="1054680">Now,</span> <span m="1054900">there's</span> <span m="1055280">two</span> <span m="1055530">cases</span> <span m="1056460">to</span> <span m="1056580">analyze</span> <span m="1057130">this,</span> <span m="1058180">based</span> <span m="1058490">on</span> <span m="1058570">what</span> <span m="1058710">happens</span> <span m="1059170">in</span> <span m="1059220">the</span> <span m="1059300">first</span> <span m="1059630">bet.</span> <span m="1060380">You</span> <span m="1060540">could</span> <span m="1060700">win</span> <span m="1060900">it,</span> <span m="1061960">or</span> <span m="1062160">you</span> <span m="1062250">could</span> <span m="1062400">lose</span> <span m="1062700">it.</span> <span m="1063600">And</span> <span m="1063710">then</span> <span m="1063840">we're</span> <span m="1063990">going</span> <span m="1064080">to</span> <span m="1064160">recurse.</span> </p>
<p><span m="1066240">So</span> <span m="1067170">we're</span> <span m="1067380">going</span> <span m="1067470">to</span> <span m="1067540">define</span> <span m="1070950">E</span> <span m="1072130">to</span> <span m="1072190">be</span> <span m="1072330">the</span> <span m="1072500">event</span> <span m="1074540">that</span> <span m="1074690">you</span> <span m="1074840">win</span> <span m="1076450">the</span> <span m="1076550">first</span> <span m="1076880">bet.</span> <span m="1080570">And</span> <span m="1080920">E</span> <span m="1081080">bar</span> <span m="1082780">is</span> <span m="1082940">the</span> <span m="1083090">event</span> <span m="1085010">that</span> <span m="1085130">you</span> <span m="1085280">lose</span> <span m="1086650">the</span> <span m="1086740">first</span> <span m="1087060">bet.</span> <span m="1092080">Now,</span> <span m="1092200">by</span> <span m="1092340">the</span> <span m="1092470">theory</span> <span m="1092860">of</span> <span m="1092950">total</span> <span m="1093290">probability,</span> <span m="1094180">which</span> <span m="1094340">we</span> <span m="1094480">did</span> <span m="1094650">in</span> <span m="1094820">recitation</span> <span m="1095890">maybe</span> <span m="1096060">a</span> <span m="1096110">couple</span> <span m="1096400">weeks</span> <span m="1096630">ago,</span> <span m="1098120">we</span> <span m="1098320">can</span> <span m="1098500">rewrite</span> <span m="1099130">this</span> <span m="1099530">depending</span> <span m="1100060">on</span> <span m="1100940">whether</span> <span m="1101210">E</span> <span m="1101430">happened</span> <span m="1101970">or</span> <span m="1102550">the</span> <span m="1102660">complement</span> <span m="1103140">of</span> <span m="1103190">E</span> <span m="1103350">happened.</span> </p>
<p><span m="1105080">And</span> <span m="1105360">you</span> <span m="1105470">get</span> <span m="1106170">that</span> <span m="1106650">the</span> <span m="1107340">probability</span> <span m="1107970">is</span> <span m="1108080">simply</span> <span m="1108370">the</span> <span m="1108470">probability</span> <span m="1109050">of</span> <span m="1109140">going</span> <span m="1109430">home</span> <span m="1109690">happy</span> <span m="1111990">and</span> <span m="1112440">winning</span> <span m="1112690">the</span> <span m="1112780">first</span> <span m="1113140">bet</span> <span m="1113760">times--</span> <span m="1115240">and</span> <span m="1115390">I've got to</span> <span m="1115470">put</span> <span m="1115600">the</span> <span m="1115680">conditioning</span> <span m="1116200">in.</span> <span m="1116510">That</span> <span m="1116850">doesn't</span> <span m="1117100">go</span> <span m="1117270">away.</span> <span m="1124080">So</span> <span m="1124190">I'm</span> <span m="1124270">breaking</span> <span m="1124600">into</span> <span m="1124760">two</span> <span m="1124970">cases.</span> <span m="1125510">The</span> <span m="1125610">first</span> <span m="1125810">one</span> <span m="1125950">is</span> <span m="1126070">you</span> <span m="1126200">win</span> <span m="1126370">the</span> <span m="1126470">first</span> <span m="1126770">bet</span> <span m="1128070">given</span> <span m="1129180">D</span> <span m="1129230">equals</span> <span m="1129640">n,</span> <span m="1131350">And</span> <span m="1131630">the</span> <span m="1131730">case</span> <span m="1132140">where</span> <span m="1132260">you</span> <span m="1132400">lose</span> <span m="1132680">the</span> <span m="1132780">first</span> <span m="1133100">bet,</span> <span m="1138500">given</span> <span m="1138760">D</span> <span m="1138970">equals</span> <span m="1139310">n.</span> <span m="1142070">Any</span> <span m="1142250">questions</span> <span m="1142740">here?</span> </p>
<p><span m="1144601">The</span> <span m="1145090">probability</span> <span m="1145580">of</span> <span m="1145640">going</span> <span m="1145880">home</span> <span m="1146100">happy</span> <span m="1146530">given</span> <span m="1146800">you</span> <span m="1146920">start</span> <span m="1147190">with</span> <span m="1147310">n</span> <span m="1147460">dollars</span> <span m="1148640">is</span> <span m="1149060">the</span> <span m="1149150">probability</span> <span m="1149660">of</span> <span m="1149710">going</span> <span m="1149940">home</span> <span m="1150180">happy</span> <span m="1150530">and</span> <span m="1150660">winning</span> <span m="1150930">the</span> <span m="1151010">first</span> <span m="1151340">bet</span> <span m="1156310">given</span> <span m="1156620">D</span> <span m="1156790">equals</span> <span m="1157050">n</span> <span m="1158000">plus</span> <span m="1158290">the</span> <span m="1158370">probability</span> <span m="1158800">of</span> <span m="1158870">going</span> <span m="1159120">home</span> <span m="1159280">happy and</span> <span m="1159650">losing</span> <span m="1160030">the</span> <span m="1160120">first</span> <span m="1160420">bet</span> <span m="1161180">given</span> <span m="1161420">D</span> <span m="1161610">equals</span> <span m="1161900">n--</span> <span m="1162800">just</span> <span m="1163000">those</span> <span m="1163210">are</span> <span m="1163230">the</span> <span m="1163330">two</span> <span m="1163490">cases.</span> <span m="1165630">Now</span> <span m="1165840">I</span> <span m="1165910">can</span> <span m="1166090">use</span> <span m="1166340">the</span> <span m="1166420">definition</span> <span m="1166950">of</span> <span m="1167020">conditional</span> <span m="1167420">probability</span> <span m="1169290">to</span> <span m="1169430">rewrite</span> <span m="1169840">these.</span> <span m="1171256">This</span> <span m="1171730">is</span> <span m="1171990">the</span> <span m="1172080">probability--</span> <span m="1173520">you've got</span> <span m="1173720">two</span> <span m="1173880">events--</span> <span m="1174350">that</span> <span m="1174480">the</span> <span m="1174570">first</span> <span m="1174890">one</span> <span m="1175000">happens</span> <span m="1176980">given</span> <span m="1177220">D</span> <span m="1177450">equals</span> <span m="1177730">n</span> <span m="1179520">times</span> <span m="1180190">the</span> <span m="1180280">probability</span> <span m="1181410">the</span> <span m="1181510">second</span> <span m="1181760">one</span> <span m="1181930">happens</span> <span m="1183440">given</span> <span m="1183730">that</span> <span m="1183860">the</span> <span m="1183950">first</span> <span m="1184280">one</span> <span m="1184430">happened</span> <span m="1185940">and</span> <span m="1186210">D</span> <span m="1186370">equals</span> <span m="1186690">n.</span> </p>
<p><span m="1189180">This</span> <span m="1189440">is</span> <span m="1189600">just</span> <span m="1189960">the</span> <span m="1190050">definition</span> <span m="1190530">of</span> <span m="1190590">conditional</span> <span m="1190990">probability,</span> <span m="1191560">when</span> <span m="1191680">I've</span> <span m="1191750">got</span> <span m="1191930">an</span> <span m="1192040">intersection</span> <span m="1192530">of</span> <span m="1192610">events</span> <span m="1192960">here.</span> <span m="1194180">The</span> <span m="1194260">probability</span> <span m="1194670">of</span> <span m="1194740">both</span> <span m="1195050">happening</span> <span m="1195870">is</span> <span m="1196020">the</span> <span m="1196080">probability</span> <span m="1196480">of the</span> <span m="1196570">first</span> <span m="1196910">happening</span> <span m="1197760">times</span> <span m="1198060">the</span> <span m="1198120">probability</span> <span m="1198420">of</span> <span m="1198540">the</span> <span m="1198630">second</span> <span m="1199000">happening</span> <span m="1199390">given</span> <span m="1199600">that</span> <span m="1199690">the</span> <span m="1199770">first</span> <span m="1200080">happened.</span> <span m="1200960">And</span> <span m="1201070">of</span> <span m="1201120">course,</span> <span m="1201440">everything</span> <span m="1201900">is</span> <span m="1202010">in</span> <span m="1202080">this</span> <span m="1202240">universe</span> <span m="1202595">of D</span> <span m="1202950">equals</span> <span m="1203310">n.</span> <span m="1205570">So I've</span> <span m="1205720">used</span> <span m="1206080">it</span> <span m="1206210">in a</span> <span m="1206280">little</span> <span m="1206920">different</span> <span m="1207310">twist</span> <span m="1207670">than</span> <span m="1207830">we</span> <span m="1207920">had</span> <span m="1208120">it</span> <span m="1208160">before.</span> </p>
<p><span m="1209490">The</span> <span m="1209540">same</span> <span m="1209790">thing</span> <span m="1209950">over</span> <span m="1210160">here--</span> <span m="1210650">this</span> <span m="1210930">now</span> <span m="1211240">is</span> <span m="1211530">the</span> <span m="1211610">probability</span> <span m="1213010">of</span> <span m="1213130">E</span> <span m="1213310">prime</span> <span m="1214710">given</span> <span m="1214930">D</span> <span m="1215300">equals</span> <span m="1215720">n</span> <span m="1217140">times</span> <span m="1217510">the</span> <span m="1217590">probability</span> <span m="1218290">of</span> <span m="1218420">W</span> <span m="1218750">star--</span> <span m="1219610">winning,</span> <span m="1220200">going</span> <span m="1220390">home</span> <span m="1220540">happy--</span> <span m="1221670">given</span> <span m="1223984">that</span> <span m="1224400">you</span> <span m="1224540">lost</span> <span m="1224900">the</span> <span m="1224990">first</span> <span m="1225360">bet</span> <span m="1226000">and</span> <span m="1226330">D</span> <span m="1226600">equals</span> <span m="1226950">n.</span> <span m="1229420">That's</span> <span m="1229670">D</span> <span m="1229840">equals</span> <span m="1230130">n</span> <span m="1230290">there.</span> <span m="1232370">So it</span> <span m="1232500">looks</span> <span m="1232740">like</span> <span m="1232840">it's</span> <span m="1232970">got</span> <span m="1233180">more</span> <span m="1233390">complicated,</span> <span m="1234540">but</span> <span m="1234840">now</span> <span m="1235060">we</span> <span m="1235160">can</span> <span m="1235270">start</span> <span m="1235620">simplifying.</span> </p>
<p><span m="1238650">What's</span> <span m="1238900">the</span> <span m="1238990">probability</span> <span m="1240110">of</span> <span m="1240260">winning</span> <span m="1240610">the</span> <span m="1240720">first</span> <span m="1241060">bet</span> <span m="1241460">given</span> <span m="1241690">that you</span> <span m="1241790">started</span> <span m="1242280">with</span> <span m="1242440">n</span> <span m="1242580">dollars?</span> </p>
<p><span m="1244022">AUDIENCE: p.</span> </p>
<p><span m="1245390">PROFESSOR: p--</span> <span m="1247100">in</span> <span m="1247210">fact,</span> <span m="1247530">does</span> <span m="1247610">this</span> <span m="1247780">have</span> <span m="1247950">anything</span> <span m="1248340">to</span> <span m="1248440">do</span> <span m="1248720">with</span> <span m="1248830">the</span> <span m="1248910">probability</span> <span m="1249410">of</span> <span m="1249480">winning</span> <span m="1249690">the</span> <span m="1249770">first</span> <span m="1250070">bet?</span> <span m="1251730">No,</span> <span m="1252840">this</span> <span m="1253050">is</span> <span m="1253180">just</span> <span m="1253540">p.</span> <span m="1259610">Now,</span> <span m="1260210">what</span> <span m="1260380">about</span> <span m="1260650">this</span> <span m="1260940">thing?</span> </p>
<p><span m="1263290">I</span> <span m="1263470">am</span> <span m="1263710">conditioning</span> <span m="1264490">on</span> <span m="1264820">winning</span> <span m="1265990">the</span> <span m="1266110">first</span> <span m="1266490">bet</span> <span m="1272000">given</span> <span m="1273620">and</span> <span m="1274250">I</span> <span m="1274410">start</span> <span m="1274720">with</span> <span m="1274850">n</span> <span m="1275030">dollars.</span> <span m="1279750">What's</span> <span m="1280000">another</span> <span m="1280280">way</span> <span m="1280560">of</span> <span m="1281220">expressing</span> <span m="1281860">I</span> <span m="1281930">won</span> <span m="1282220">the</span> <span m="1282310">first</span> <span m="1282650">bet</span> <span m="1282930">and</span> <span m="1283050">I</span> <span m="1283120">started</span> <span m="1283520">with</span> <span m="1283680">n</span> <span m="1283830">dollars?</span> <span m="1285310">Yeah?</span> </p>
<p><span m="1286087">AUDIENCE: You have</span> <span m="1286564">n plus $1.</span> </p>
<p><span m="1287520">PROFESSOR: I</span> <span m="1287640">now</span> <span m="1287970">have</span> <span m="1288170">n</span> <span m="1288330">plus</span> <span m="1288590">$1</span> <span m="1289900">going</span> <span m="1290180">forward.</span> <span m="1291860">And</span> <span m="1292650">because</span> <span m="1293220">I</span> <span m="1293270">have</span> <span m="1293380">a</span> <span m="1293430">martingale,</span> <span m="1294360">and</span> <span m="1294500">everything</span> <span m="1294900">is</span> <span m="1295020">mutually</span> <span m="1295450">independent,</span> <span m="1296445">it's</span> <span m="1296710">like</span> <span m="1296880">the</span> <span m="1296970">world</span> <span m="1297340">starts</span> <span m="1297750">all</span> <span m="1297980">over</span> <span m="1298180">again.</span> <span m="1298700">I'm</span> <span m="1298800">now</span> <span m="1299070">in</span> <span m="1299160">a</span> <span m="1299220">state</span> <span m="1299680">with</span> <span m="1299830">n</span> <span m="1300050">plus</span> <span m="1300300">$1,</span> <span m="1301440">and</span> <span m="1301670">I</span> <span m="1301720">want</span> <span m="1301980">to</span> <span m="1302030">know</span> <span m="1302170">the</span> <span m="1302280">probability</span> <span m="1304110">that</span> <span m="1304230">I</span> <span m="1304290">go</span> <span m="1304480">home</span> <span m="1304710">happy.</span> </p>
<p><span m="1305375">It</span> <span m="1305640">doesn't</span> <span m="1306000">matter</span> <span m="1306350">how</span> <span m="1306790">I</span> <span m="1306900">got</span> <span m="1307090">the</span> <span m="1307220">n</span> <span m="1307340">plus</span> <span m="1307580">$1.</span> <span m="1309000">It's</span> <span m="1309130">just</span> <span m="1309370">going</span> <span m="1309670">forward--</span> <span m="1310230">I</span> <span m="1310290">got</span> <span m="1310480">n</span> <span m="1310590">plus</span> <span m="1310770">$1</span> <span m="1311210">in my</span> <span m="1311340">pocket,</span> <span m="1312000">I</span> <span m="1312090">want to</span> <span m="1312300">know</span> <span m="1312340">the</span> <span m="1312420">probability</span> <span m="1312860">of going</span> <span m="1313090">home</span> <span m="1313280">happy.</span> <span m="1314270">So</span> <span m="1314300">I</span> <span m="1314390">reset</span> <span m="1315300">to</span> <span m="1315740">D</span> <span m="1316090">equals</span> <span m="1316360">n</span> <span m="1316460">plus</span> <span m="1316690">1.</span> <span m="1319740">So</span> <span m="1319860">I</span> <span m="1319950">replace</span> <span m="1320520">this</span> <span m="1321540">with</span> <span m="1321850">that,</span> <span m="1322260">because</span> <span m="1323240">however</span> <span m="1323610">long</span> <span m="1323860">it</span> <span m="1323950">took</span> <span m="1324070">me</span> <span m="1324170">to</span> <span m="1324240">get</span> <span m="1324470">there</span> <span m="1324780">and</span> <span m="1324840">all</span> <span m="1325060">that</span> <span m="1325230">stuff</span> <span m="1325530">doesn't</span> <span m="1325850">matter</span> <span m="1326270">for</span> <span m="1326380">this</span> <span m="1326650">analysis.</span> <span m="1327660">It's</span> <span m="1327940">all</span> <span m="1328150">mutually</span> <span m="1328560">dependent.</span> </p>
<p><span m="1333110">Probability</span> <span m="1333560">of</span> <span m="1333640">losing</span> <span m="1334190">the</span> <span m="1334280">first</span> <span m="1334620">bet</span> <span m="1335490">given</span> <span m="1335760">that</span> <span m="1335880">I</span> <span m="1335950">started</span> <span m="1336360">with</span> <span m="1336470">n</span> <span m="1336670">dollars--</span> <span m="1339150">1</span> <span m="1339400">minus</span> <span m="1339790">p.</span> <span m="1340490">Doesn't</span> <span m="1340740">matter</span> <span m="1341010">how</span> <span m="1341170">much</span> <span m="1341430">I</span> <span m="1341490">started</span> <span m="1341830">with.</span> <span m="1343290">And</span> <span m="1343450">here,</span> <span m="1343860">I</span> <span m="1344760">want</span> <span m="1344980">to know</span> <span m="1345040">the</span> <span m="1345150">probability</span> <span m="1345730">of</span> <span m="1345790">going</span> <span m="1346030">home</span> <span m="1346250">happy</span> <span m="1349120">given--</span> <span m="1350010">well,</span> <span m="1350670">if</span> <span m="1350820">I</span> <span m="1350890">lost</span> <span m="1351280">the</span> <span m="1351380">first</span> <span m="1351720">bet</span> <span m="1351950">and</span> <span m="1352080">I</span> <span m="1352140">started</span> <span m="1352560">with</span> <span m="1352720">n,</span> <span m="1353780">what</span> <span m="1353930">have</span> <span m="1354030">I</span> <span m="1354100">got?</span> <span m="1356080">n</span> <span m="1356360">minus</span> <span m="1356710">1.</span> </p>
<p><span m="1360530">It</span> <span m="1360630">doesn't</span> <span m="1360850">matter</span> <span m="1361100">how</span> <span m="1361310">I</span> <span m="1361380">got</span> <span m="1361580">to</span> <span m="1361650">n</span> <span m="1361830">minus</span> <span m="1362170">1.</span> <span m="1365000">Now</span> <span m="1365230">this</span> <span m="1365370">is</span> <span m="1365450">going</span> <span m="1365580">to</span> <span m="1365620">get</span> <span m="1365780">really</span> <span m="1366050">simple.</span> <span m="1369250">What's</span> <span m="1369540">another</span> <span m="1369820">name</span> <span m="1370070">for</span> <span m="1370170">that</span> <span m="1370370">expression?</span> <span m="1372720">X n</span> <span m="1373130">plus</span> <span m="1373420">1.</span> </p>
<p><span m="1377560">And</span> <span m="1377780">another</span> <span m="1378040">name</span> <span m="1378290">for</span> <span m="1378380">this</span> <span m="1378630">expression?</span> <span m="1380800">X</span> <span m="1381100">n</span> <span m="1381280">minus</span> <span m="1381660">1.</span> <span m="1385680">So</span> <span m="1385970">we</span> <span m="1386130">proved</span> <span m="1386900">that</span> <span m="1387820">X</span> <span m="1388190">n</span> <span m="1388970">equals</span> <span m="1389890">p</span> <span m="1390130">X n</span> <span m="1390480">plus</span> <span m="1390730">1</span> <span m="1390980">plus</span> <span m="1391220">1</span> <span m="1391400">minus</span> <span m="1391720">p X</span> <span m="1392140">n</span> <span m="1392260">minus</span> <span m="1392610">1.</span> <span m="1392770">And</span> <span m="1392850">that's</span> <span m="1393070">what</span> <span m="1393180">I</span> <span m="1393230">claimed</span> <span m="1393580">is</span> <span m="1393720">true.</span> <span m="1395100">So</span> <span m="1395300">we</span> <span m="1395430">finished</span> <span m="1395750">the</span> <span m="1395870">proof.</span> <span m="1398380">Any</span> <span m="1398500">questions?</span> </p>
<p><span m="1400318">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1401130">PROFESSOR: Did i</span> <span m="1401220">screw it</span> <span m="1401560">up?</span> </p>
<p><span m="1402416">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1403272">PROFESSOR: I</span> <span m="1403700">claim</span> <span m="1407100">probability</span> <span m="1407690">of</span> <span m="1407770">winning--</span> <span m="1409540">so</span> <span m="1409730">let's</span> <span m="1409940">see</span> <span m="1410100">if</span> <span m="1410450">I</span> <span m="1410520">have</span> <span m="1410670">a</span> <span m="1410770">wrong</span> <span m="1411080">in</span> <span m="1411180">here.</span> <span m="1411590">I might</span> <span m="1411720">have</span> <span m="1412110">screwed</span> <span m="1412540">it up.</span> <span m="1412950">I think</span> <span m="1413130">I</span> <span m="1413190">proved</span> <span m="1413590">it's</span> <span m="1414900">n</span> <span m="1415110">plus</span> <span m="1415520">1,</span> <span m="1415840">right?</span> </p>
<p><span m="1418440">Yep,</span> <span m="1418700">sure</span> <span m="1418940">enough,</span> <span m="1419310">I</span> <span m="1419380">think</span> <span m="1419550">this</span> <span m="1419720">is</span> <span m="1419810">a</span> <span m="1419870">plus</span> <span m="1420180">1.</span> <span m="1421230">That's</span> <span m="1421450">a</span> <span m="1421500">minus</span> <span m="1421880">1.</span> <span m="1423210">Now,</span> <span m="1423500">it's always</span> <span m="1423610">good</span> <span m="1423780">to</span> <span m="1423860">check</span> <span m="1424130">to</span> <span m="1424250">you</span> <span m="1424400">proved</span> <span m="1424810">what</span> <span m="1425620">you</span> <span m="1425720">said</span> <span m="1425960">you were</span> <span m="1426100">going</span> <span m="1426200">to</span> <span m="1426260">prove.</span> </p>
<p><span m="1426900">So</span> <span m="1427690">I</span> <span m="1427820">needed</span> <span m="1428090">to</span> <span m="1428220">change</span> <span m="1428550">this.</span> <span m="1430230">That's</span> <span m="1430310">what</span> <span m="1430410">I</span> <span m="1430480">proved.</span> <span m="1437290">Any</span> <span m="1437490">other</span> <span m="1437640">questions?</span> <span m="1438480">That</span> <span m="1438580">was</span> <span m="1438700">a</span> <span m="1438760">pretty</span> <span m="1438950">important</span> <span m="1439330">question.</span> </p>
<p><span m="1442370">All</span> <span m="1442520">right,</span> <span m="1442820">so</span> <span m="1444740">we</span> <span m="1444890">have</span> <span m="1445040">a</span> <span m="1445120">recurrence</span> <span m="1446600">for</span> <span m="1446810">X</span> <span m="1447070">n.</span> <span m="1447620">Now,</span> <span m="1448690">it's</span> <span m="1448880">a</span> <span m="1448930">little</span> <span m="1449260">funny</span> <span m="1449520">looking</span> <span m="1449820">at</span> <span m="1449950">first,</span> <span m="1450430">because</span> <span m="1451140">normally</span> <span m="1451520">with</span> <span m="1451650">a</span> <span m="1451690">recurrence,</span> <span m="1452430">X n</span> <span m="1452820">would</span> <span m="1452960">depend</span> <span m="1453340">on</span> <span m="1454230">X</span> <span m="1454540">sub</span> <span m="1454730">i</span> <span m="1454930">that</span> <span m="1455020">are</span> <span m="1455130">smaller--</span> <span m="1455770">the</span> <span m="1455960">i's</span> <span m="1456260">are</span> <span m="1456350">smaller</span> <span m="1456720">than</span> <span m="1456870">n.</span> <span m="1458040">So</span> <span m="1458180">it</span> <span m="1458220">looks</span> <span m="1458410">a</span> <span m="1458460">little</span> <span m="1458730">wacky.</span> </p>
<p><span m="1460210">But</span> <span m="1460840">is</span> <span m="1460980">that</span> <span m="1461120">a</span> <span m="1461170">problem?</span> <span m="1464320">I</span> <span m="1464460">can</span> <span m="1464600">just</span> <span m="1464750">solve</span> <span m="1465080">for</span> <span m="1465190">X n</span> <span m="1465500">plus</span> <span m="1465690">1--</span> <span m="1466550">just</span> <span m="1466750">subtract</span> <span m="1467280">this</span> <span m="1467480">and</span> <span m="1467580">put</span> <span m="1467720">it</span> <span m="1467810">over</span> <span m="1467960">there.</span> <span m="1469550">So</span> <span m="1469660">let's</span> <span m="1470050">do</span> <span m="1470190">that.</span> </p>
<p></p>
<p><span m="1487740">OK,</span> <span m="1488150">so</span> <span m="1488690">if</span> <span m="1488860">I</span> <span m="1490450">solve</span> <span m="1490960">for</span> <span m="1491190">X</span> <span m="1491455">n</span> <span m="1491720">plus</span> <span m="1492030">1</span> <span m="1492250">up</span> <span m="1492390">there,</span> <span m="1492490">I'll</span> <span m="1492610">put</span> <span m="1492830">p</span> <span m="1493105">X n</span> <span m="1493380">plus</span> <span m="1493580">1</span> <span m="1493820">on its</span> <span m="1493890">own</span> <span m="1494130">side,</span> <span m="1495280">I</span> <span m="1495390">get</span> <span m="1495630">p</span> <span m="1497070">X n</span> <span m="1497550">plus</span> <span m="1497870">1</span> <span m="1499840">minus</span> <span m="1500510">X</span> <span m="1500800">n</span> <span m="1502390">plus</span> <span m="1502930">1</span> <span m="1503210">minus</span> <span m="1503600">p</span> <span m="1505380">X</span> <span m="1505690">n</span> <span m="1505830">minus</span> <span m="1506240">1</span> <span m="1507140">equals</span> <span m="1507520">0.</span> <span m="1508690">And</span> <span m="1508850">I</span> <span m="1508910">know</span> <span m="1509100">that</span> <span m="1509260">X</span> <span m="1509530">0</span> <span m="1509910">is</span> <span m="1510080">0.</span> <span m="1511260">And</span> <span m="1511400">I</span> <span m="1511440">know</span> <span m="1511610">that</span> <span m="1511760">X</span> <span m="1512160">T</span> <span m="1512560">equals</span> <span m="1512980">1.</span> </p>
<p><span m="1517320">Now,</span> <span m="1517510">what</span> <span m="1517670">type</span> <span m="1517850">of</span> <span m="1517950">recurrence</span> <span m="1518470">is</span> <span m="1518580">this?</span> </p>
<p><span m="1521860">AUDIENCE: Linear.</span> </p>
<p><span m="1522770">PROFESSOR: Linear,</span> <span m="1523570">good,</span> <span m="1524480">so</span> <span m="1524580">it's</span> <span m="1524690">a</span> <span m="1524750">linear</span> <span m="1525080">recurrence.</span> <span m="1526130">And</span> <span m="1526450">what</span> <span m="1526720">type</span> <span m="1527040">of</span> <span m="1527170">linear</span> <span m="1527580">recurrence</span> <span m="1528160">is</span> <span m="1528260">it?</span> </p>
<p><span m="1528853">AUDIENCE: Homogeneous.</span> </p>
<p><span m="1529800">PROFESSOR: Homogeneous--</span> <span m="1531100">that's</span> <span m="1531370">the</span> <span m="1531640">best</span> <span m="1531910">case,</span> <span m="1532540">simple</span> <span m="1532890">case,</span> <span m="1533200">that's</span> <span m="1533440">good.</span> <span m="1534110">The</span> <span m="1534220">boundary</span> <span m="1534630">conditions</span> <span m="1535100">are</span> <span m="1535200">a</span> <span m="1535260">little</span> <span m="1535550">weird,</span> <span m="1535830">because</span> <span m="1536610">the</span> <span m="1536740">recurrences</span> <span m="1537230">we</span> <span m="1537330">all</span> <span m="1537470">saw</span> <span m="1537640">before,</span> <span m="1538870">if</span> <span m="1538940">we</span> <span m="1539030">had</span> <span m="1539250">two</span> <span m="1539450">boundary</span> <span m="1539790">conditions</span> <span m="1540270">it would</span> <span m="1540370">be</span> <span m="1540480">X0</span> <span m="1540930">and</span> <span m="1541010">X1.</span> <span m="1541870">Here</span> <span m="1542070">it's</span> <span m="1542190">X0</span> <span m="1542400">and</span> <span m="1542660">X T.</span> </p>
<p><span m="1543260">But</span> <span m="1543920">all's</span> <span m="1544130">you</span> <span m="1544210">need</span> <span m="1544370">are</span> <span m="1544430">two.</span> <span m="1545330">Doesn't</span> <span m="1545580">matter</span> <span m="1545830">where</span> <span m="1546000">they</span> <span m="1546150">are.</span> <span m="1548960">So</span> <span m="1549080">how</span> <span m="1549250">do</span> <span m="1549350">I</span> <span m="1549450">solve</span> <span m="1549800">that</span> <span m="1550000">thing?</span> <span m="1550220">What's</span> <span m="1550430">the</span> <span m="1550510">next</span> <span m="1550780">thing</span> <span m="1550960">I</span> <span m="1551390">do?</span> <span m="1554070">What</span> <span m="1554260">is</span> <span m="1554380">it?</span> </p>
<p><span m="1555220">AUDIENCE: Characterize</span> <span m="1555660">the equation.</span> </p>
<p><span m="1556540">PROFESSOR: Characterize</span> <span m="1556805">the</span> <span m="1557070">equation.</span> <span m="1557560">And</span> <span m="1557640">what</span> <span m="1557760">do</span> <span m="1557800">you</span> <span m="1557890">do</span> <span m="1558050">it</span> <span m="1558110">that</span> <span m="1558250">equation?</span> </p>
<p><span m="1559420">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="1560210">PROFESSOR: Solve</span> <span m="1560660">it,</span> <span m="1560840">get</span> <span m="1560950">the</span> <span m="1561060">roots.</span> <span m="1561520">This'll</span> <span m="1561740">be</span> <span m="1561850">good</span> <span m="1562010">practice</span> <span m="1562530">for</span> <span m="1562650">the</span> <span m="1563470">final,</span> <span m="1563820">because</span> <span m="1564020">you'll</span> <span m="1564120">probably</span> <span m="1564460">have</span> <span m="1564640">to</span> <span m="1564760">do</span> <span m="1564910">something</span> <span m="1565170">like</span> <span m="1565370">this.</span> <span m="1566820">So</span> <span m="1567010">that's</span> <span m="1567280">the</span> <span m="1567380">characteristic</span> <span m="1567990">equation.</span> </p>
<p><span m="1572800">And</span> <span m="1573050">what's</span> <span m="1573260">the</span> <span m="1573380">order</span> <span m="1573710">of</span> <span m="1573770">this</span> <span m="1573940">equation--</span> <span m="1575580">the</span> <span m="1575680">degree?</span> <span m="1580340">That's</span> <span m="1580640">going</span> <span m="1581970">to</span> <span m="1582040">be</span> <span m="1586210">2,</span> <span m="1586500">right?</span> <span m="1586730">I'm</span> <span m="1586810">going</span> <span m="1586890">to</span> <span m="1586940">have pr</span> <span m="1587190">squared</span> <span m="1588770">minus</span> <span m="1589200">r</span> <span m="1590120">plus</span> <span m="1590630">1</span> <span m="1590800">minus</span> <span m="1591130">p</span> <span m="1592070">is</span> <span m="1592560">0.</span> <span m="1593600">That's</span> <span m="1593870">my</span> <span m="1594010">characteristic</span> <span m="1594600">equation.</span> </p>
<p><span m="1597180">Remember</span> <span m="1597410">that?</span> <span m="1597760">So</span> <span m="1597880">I</span> <span m="1599190">make</span> <span m="1599400">this</span> <span m="1599570">be</span> <span m="1599660">the</span> <span m="1599770">constant</span> <span m="1600250">term.</span> <span m="1600640">Then</span> <span m="1600780">I</span> <span m="1600850">have</span> <span m="1601120">the</span> <span m="1601550">first-order</span> <span m="1602000">term,</span> <span m="1602300">then</span> <span m="1602420">the</span> <span m="1602490">second-order</span> <span m="1602980">term.</span> </p>
<p><span m="1605210">All</span> <span m="1605350">right,</span> <span m="1605660">now</span> <span m="1605820">I</span> <span m="1605910">solve</span> <span m="1606340">it.</span> <span m="1608570">And</span> <span m="1608640">that's</span> <span m="1608880">easy</span> <span m="1609160">for</span> <span m="1609370">a</span> <span m="1609430">second-order</span> <span m="1610320">equation.</span> <span m="1612080">1</span> <span m="1612300">plus</span> <span m="1612620">or</span> <span m="1612680">minus</span> <span m="1613180">the</span> <span m="1613260">square</span> <span m="1613700">root</span> <span m="1613980">of</span> <span m="1615190">1</span> <span m="1615850">minus</span> <span m="1617840">4p</span> <span m="1619340">1</span> <span m="1619540">minus</span> <span m="1619920">p</span> <span m="1621920">over</span> <span m="1623390">2</span> <span m="1623630">p.</span> <span m="1627140">Let's</span> <span m="1627540">do</span> <span m="1627690">that.</span> </p>
<p><span m="1630810">OK,</span> <span m="1631350">so</span> <span m="1631500">this</span> <span m="1631760">is</span> <span m="1632130">1</span> <span m="1632550">plus</span> <span m="1632860">or</span> <span m="1632950">minus</span> <span m="1634220">the</span> <span m="1634270">square</span> <span m="1634750">root</span> <span m="1635380">of</span> <span m="1635420">1</span> <span m="1635690">minus</span> <span m="1636090">4p</span> <span m="1637400">plus</span> <span m="1637800">4p</span> <span m="1638280">squared</span> <span m="1640030">over</span> <span m="1640240">2p.</span> <span m="1648730">Just</span> <span m="1648960">using</span> <span m="1649160">the</span> <span m="1649250">quadratic</span> <span m="1649820">formula</span> <span m="1650370">and</span> <span m="1650390">simplifying.</span> <span m="1655780">And</span> <span m="1656260">it</span> <span m="1656340">works</span> <span m="1656590">out</span> <span m="1656710">really</span> <span m="1656980">nicely,</span> <span m="1658400">because</span> <span m="1658660">that</span> <span m="1658930">is</span> <span m="1659620">the</span> <span m="1659720">square</span> <span m="1660130">root</span> <span m="1660350">of--</span> <span m="1660390">this</span> <span m="1660530">is</span> <span m="1660710">just</span> <span m="1660940">1</span> <span m="1661150">minus</span> <span m="1661640">2p</span> <span m="1662020">squared.</span> </p>
<p><span m="1668990">So</span> <span m="1669070">that's</span> <span m="1669370">1</span> <span m="1669910">plus</span> <span m="1670270">or</span> <span m="1670330">minus</span> <span m="1671300">1</span> <span m="1671480">minus</span> <span m="1671820">2p</span> <span m="1673210">over</span> <span m="1673560">2p.</span> <span m="1675740">And</span> <span m="1675890">that</span> <span m="1676140">is</span> <span m="1678340">2</span> <span m="1678680">minus</span> <span m="1679070">2p</span> <span m="1679720">over</span> <span m="1679980">2p</span> <span m="1681540">or</span> <span m="1683960">1</span> <span m="1684260">minus</span> <span m="1684700">1</span> <span m="1685650">cancels,</span> <span m="1686410">then</span> <span m="1688470">minus</span> <span m="1689500">2p</span> <span m="1690160">is</span> <span m="1690340">2p</span> <span m="1691390">over</span> <span m="1691620">2p.</span> </p>
<p><span m="1693020">So</span> <span m="1693400">the</span> <span m="1693570">answers,</span> <span m="1694110">the</span> <span m="1694240">roots</span> <span m="1694730">are</span> <span m="1696300">divide</span> <span m="1696660">by</span> <span m="1696790">2</span> <span m="1697130">on</span> <span m="1697250">this</span> <span m="1697440">one.</span> <span m="1697590">I</span> <span m="1697650">get</span> <span m="1697910">1</span> <span m="1698330">minus</span> <span m="1698710">p</span> <span m="1699650">over</span> <span m="1699920">p</span> <span m="1701030">and</span> <span m="1701280">1.</span> <span m="1703070">Those</span> <span m="1703380">are</span> <span m="1703420">the</span> <span m="1703520">roots.</span> <span m="1707640">Are</span> <span m="1707750">these</span> <span m="1707980">roots</span> <span m="1708350">different?</span> <span m="1711290">Do I have</span> <span m="1711420">the</span> <span m="1711560">case</span> <span m="1711830">of</span> <span m="1711910">a</span> <span m="1711960">double</span> <span m="1712270">root?</span> <span m="1712460">Are</span> <span m="1712520">the</span> <span m="1712650">roots</span> <span m="1713510">always</span> <span m="1713860">different?</span> </p>
<p><span m="1716510">They're</span> <span m="1716750">usually</span> <span m="1717180">different.</span> <span m="1717490">What's</span> <span m="1717700">the</span> <span m="1717810">case</span> <span m="1718130">where</span> <span m="1718220">these</span> <span m="1718460">roots</span> <span m="1718890">are</span> <span m="1719060">the</span> <span m="1719180">same?</span> </p>
<p><span m="1719880">AUDIENCE: 0.5.</span> </p>
<p><span m="1720870">PROFESSOR: 0.5,</span> <span m="1722350">which is</span> <span m="1722610">sort</span> <span m="1722850">of</span> <span m="1722930">an</span> <span m="1723000">interesting</span> <span m="1723400">case</span> <span m="1723760">in</span> <span m="1723840">this</span> <span m="1724060">game.</span> <span m="1725990">Because</span> <span m="1726260">if p equals</span> <span m="1726900">1/2,</span> <span m="1727450">we</span> <span m="1727530">have</span> <span m="1727700">an</span> <span m="1727780">unbiased</span> <span m="1728450">random</span> <span m="1728760">walk.</span> <span m="1729110">You</span> <span m="1729190">got</span> <span m="1729350">a</span> <span m="1729400">fair</span> <span m="1729830">game.</span> </p>
<p><span m="1731250">And</span> <span m="1731380">so</span> <span m="1731570">it</span> <span m="1731720">says</span> <span m="1732120">right</span> <span m="1732380">away,</span> <span m="1732650">well,</span> <span m="1732770">maybe</span> <span m="1733130">the</span> <span m="1733240">result</span> <span m="1733730">is</span> <span m="1733840">going</span> <span m="1733950">to</span> <span m="1734000">be</span> <span m="1734130">different</span> <span m="1734590">for</span> <span m="1734730">a</span> <span m="1734780">fair</span> <span m="1735130">game</span> <span m="1736270">than</span> <span m="1736540">the</span> <span m="1736620">game</span> <span m="1736860">we're</span> <span m="1736940">playing</span> <span m="1737200">in the</span> <span m="1737290">casino,</span> <span m="1738140">where</span> <span m="1738320">it's</span> <span m="1738450">biased.</span> <span m="1741300">So</span> <span m="1742060">let's</span> <span m="1742310">look</span> <span m="1742430">at</span> <span m="1742500">the</span> <span m="1742810">casino</span> <span m="1743190">game</span> <span m="1743470">where</span> <span m="1743610">p</span> <span m="1743810">is</span> <span m="1743970">not</span> <span m="1744230">1/2.</span> <span m="1744690">Then</span> <span m="1744890">the</span> <span m="1745000">roots</span> <span m="1745250">are</span> <span m="1745300">different.</span> <span m="1748280">Later,</span> <span m="1748560">we'll</span> <span m="1748710">go</span> <span m="1748860">back</span> <span m="1749120">and</span> <span m="1749240">analyze</span> <span m="1749570">the</span> <span m="1749660">case</span> <span m="1750000">when</span> <span m="1750070">the</span> <span m="1750220">roots</span> <span m="1750480">of</span> <span m="1750530">the</span> <span m="1750620">same</span> <span m="1752090">for</span> <span m="1752210">the</span> <span m="1752310">fair</span> <span m="1752570">game.</span> </p>
<p></p>
<p><span m="1760310">So</span> <span m="1760460">if</span> <span m="1760720">p</span> <span m="1760990">is</span> <span m="1761150">not</span> <span m="1761380">1/2,</span> <span m="1766910">then</span> <span m="1767180">we</span> <span m="1767280">can</span> <span m="1767420">solve</span> <span m="1767900">for</span> <span m="1768080">X</span> <span m="1768300">n.</span> <span m="1770250">X n</span> <span m="1770880">is</span> <span m="1770990">some</span> <span m="1771240">constant</span> <span m="1773440">times</span> <span m="1773760">the</span> <span m="1773860">first</span> <span m="1774230">root</span> <span m="1775670">to</span> <span m="1775760">the</span> <span m="1775900">nth</span> <span m="1776100">power</span> <span m="1777960">plus</span> <span m="1778300">a</span> <span m="1778370">constant</span> <span m="1779150">times</span> <span m="1779460">the</span> <span m="1779550">second</span> <span m="1780000">root</span> <span m="1780390">to</span> <span m="1780560">the</span> <span m="1780700">nth</span> <span m="1780890">power.</span> <span m="1782180">Remember,</span> <span m="1782370">that's</span> <span m="1782620">how</span> <span m="1782770">it</span> <span m="1782820">works</span> <span m="1783110">for</span> <span m="1783300">any</span> <span m="1783590">linear</span> <span m="1784740">homogeneous</span> <span m="1785690">recurrence.</span> </p>
<p><span m="1789300">And</span> <span m="1789830">that's</span> <span m="1790080">easy,</span> <span m="1791670">because</span> <span m="1791890">the</span> <span m="1791980">second</span> <span m="1792320">root</span> <span m="1792500">was</span> <span m="1792650">1.</span> <span m="1793900">This</span> <span m="1794080">is</span> <span m="1794160">just</span> <span m="1794380">plus</span> <span m="1794640">B.</span> <span m="1795750">1 to</span> <span m="1795790">the</span> <span m="1795950">n</span> <span m="1796130">is</span> <span m="1796830">1.</span> </p>
<p><span m="1799960">How</span> <span m="1800200">do</span> <span m="1800320">I</span> <span m="1800410">figure</span> <span m="1800790">out</span> <span m="1800900">what</span> <span m="1801030">A and</span> <span m="1801310">B</span> <span m="1801530">are?</span> </p>
<p><span m="1803320">AUDIENCE: Boundary conditions.</span> </p>
<p><span m="1804300">PROFESSOR: Boundary</span> <span m="1804710">conditions,</span> <span m="1805320">very</span> <span m="1805600">good.</span> <span m="1807090">So</span> <span m="1807720">let's</span> <span m="1807920">look</span> <span m="1808040">at</span> <span m="1808110">the</span> <span m="1808190">boundary</span> <span m="1808510">conditions.</span> </p>
<p></p>
<p><span m="1830250">OK,</span> <span m="1830650">so</span> <span m="1831330">the</span> <span m="1831380">first</span> <span m="1831700">boundary</span> <span m="1832030">condition</span> <span m="1832580">is</span> <span m="1832870">at</span> <span m="1833300">0.</span> <span m="1836100">So</span> <span m="1836430">we</span> <span m="1836560">have</span> <span m="1836970">0</span> <span m="1837730">equals</span> <span m="1838330">X 0.</span> <span m="1840410">Plugging</span> <span m="1840755">in</span> <span m="1841100">there--</span> <span m="1841510">oops</span> <span m="1841680">I</span> <span m="1841710">forgot</span> <span m="1842000">the</span> <span m="1842160">n</span> <span m="1842440">up</span> <span m="1842690">here.</span> <span m="1844570">Plugging in</span> <span m="1845030">n</span> <span m="1845180">equals</span> <span m="1845500">0--</span> <span m="1846580">well,</span> <span m="1846750">this</span> <span m="1846970">to</span> <span m="1847060">the</span> <span m="1847150">0</span> <span m="1847510">is just</span> <span m="1847810">1.</span> <span m="1848140">That</span> <span m="1848410">is</span> <span m="1848600">A</span> <span m="1848720">plus</span> <span m="1849000">B.</span> <span m="1851080">That</span> <span m="1851270">means</span> <span m="1851640">that</span> <span m="1852670">B</span> <span m="1853670">equals</span> <span m="1854000">minus</span> <span m="1854430">A.</span> </p>
<p><span m="1857060">Then</span> <span m="1857180">the</span> <span m="1857270">second</span> <span m="1857630">boundary</span> <span m="1857980">condition</span> <span m="1858470">is</span> <span m="1858580">1</span> <span m="1858840">equals</span> <span m="1859980">X</span> <span m="1860250">sub</span> <span m="1860430">T.</span> <span m="1862490">And</span> <span m="1862660">that</span> <span m="1862900">is</span> <span m="1864510">A</span> <span m="1865160">1</span> <span m="1865420">minus</span> <span m="1865880">p</span> <span m="1866260">over</span> <span m="1866500">p</span> <span m="1866990">to</span> <span m="1867100">the</span> <span m="1867240">T</span> <span m="1869020">plus</span> <span m="1869300">B,</span> <span m="1869780">but</span> <span m="1869800">B</span> <span m="1870000">was</span> <span m="1870160">minus</span> <span m="1870580">A.</span> <span m="1873442">And</span> <span m="1873920">now</span> <span m="1874120">I</span> <span m="1874220">can</span> <span m="1874380">solve</span> <span m="1874870">for</span> <span m="1875110">A.</span> </p>
<p><span m="1876860">So</span> <span m="1877650">that</span> <span m="1877910">means</span> <span m="1878310">that</span> <span m="1879736">A</span> <span m="1882950">equals</span> <span m="1883710">1</span> <span m="1884130">over</span> <span m="1886930">1</span> <span m="1887270">minus</span> <span m="1887660">p</span> <span m="1888170">over</span> <span m="1888430">p</span> <span m="1888850">to</span> <span m="1888930">the</span> <span m="1889080">T</span> <span m="1890220">minus</span> <span m="1890580">1.</span> <span m="1892200">And</span> <span m="1892550">B</span> <span m="1892830">is</span> <span m="1893000">negative</span> <span m="1893420">A--</span> <span m="1894830">minus</span> <span m="1895250">1</span> <span m="1896250">over</span> <span m="1896640">1</span> <span m="1896900">minus</span> <span m="1897310">p,</span> <span m="1898610">over</span> <span m="1898670">p</span> <span m="1899120">to the</span> <span m="1899570">T</span> <span m="1900020">minus</span> <span m="1900420">1.</span> <span m="1903540">And</span> <span m="1904930">then</span> <span m="1905130">I</span> <span m="1905200">plug</span> <span m="1905690">those</span> <span m="1906120">back</span> <span m="1906450">in</span> <span m="1906920">to</span> <span m="1907000">the</span> <span m="1907100">formula</span> <span m="1908520">for</span> <span m="1908750">X n.</span> </p>
<p><span m="1911140">So</span> <span m="1911770">here's</span> <span m="1912070">my</span> <span m="1912200">constant</span> <span m="1912460">A.</span> <span m="1912720">I</span> <span m="1913110">multiply</span> <span m="1913510">that</span> <span m="1913840">times</span> <span m="1914140">1</span> <span m="1914320">minus</span> <span m="1914610">p</span> <span m="1914730">over</span> <span m="1914980">p</span> <span m="1915190">to</span> <span m="1915250">the</span> <span m="1915420">n,</span> <span m="1915850">plus</span> <span m="1916750">I</span> <span m="1916870">add</span> <span m="1917160">this</span> <span m="1917390">in.</span> <span m="1919020">So</span> <span m="1919180">this</span> <span m="1919460">means</span> <span m="1919850">that</span> <span m="1921460">the</span> <span m="1921540">probability</span> <span m="1922140">of</span> <span m="1922210">going</span> <span m="1922430">home</span> <span m="1922620">a</span> <span m="1922700">winner</span> <span m="1924430">is</span> <span m="1924570">1</span> <span m="1924780">minus</span> <span m="1925130">p</span> <span m="1925450">over</span> <span m="1925650">p</span> <span m="1926650">to</span> <span m="1926790">the</span> <span m="1926980">n</span> <span m="1928710">over</span> <span m="1928910">that</span> <span m="1929190">thing--</span> <span m="1929960">1</span> <span m="1930150">minus</span> <span m="1930500">p</span> <span m="1931530">over</span> <span m="1931650">p</span> <span m="1932010">to</span> <span m="1932090">the</span> <span m="1932260">T</span> <span m="1932650">minus</span> <span m="1933050">1,</span> <span m="1934000">plus</span> <span m="1934480">the</span> <span m="1934580">B</span> <span m="1934830">term,</span> <span m="1935260">which</span> <span m="1935460">really</span> <span m="1935680">is</span> <span m="1935810">a</span> <span m="1935860">minus</span> <span m="1936770">term</span> <span m="1937080">here,</span> <span m="1938120">is</span> <span m="1939530">just</span> <span m="1939740">minus</span> <span m="1941390">1.</span> <span m="1942910">Put</span> <span m="1943070">that</span> <span m="1943220">on</span> <span m="1943360">top</span> <span m="1943700">here.</span> </p>
<p><span m="1946620">That</span> <span m="1946950">sort</span> <span m="1947170">of</span> <span m="1947230">looks</span> <span m="1947490">messy,</span> <span m="1950140">but</span> <span m="1950190">there's</span> <span m="1950350">a</span> <span m="1950420">simplification</span> <span m="1951390">to</span> <span m="1951460">get</span> <span m="1951840">an</span> <span m="1951920">upper</span> <span m="1952110">bound</span> <span m="1952430">that's</span> <span m="1952600">very</span> <span m="1953100">close.</span> <span m="1954780">In</span> <span m="1954940">particular,</span> <span m="1955730">if</span> <span m="1955840">you</span> <span m="1955950">have</span> <span m="1956250">a</span> <span m="1957000">biased</span> <span m="1957500">game</span> <span m="1957730">against</span> <span m="1958190">you--</span> <span m="1958870">so</span> <span m="1959180">if</span> <span m="1959430">p</span> <span m="1959700">is</span> <span m="1959830">less</span> <span m="1960040">than</span> <span m="1960240">1/2,</span> <span m="1960680">as</span> <span m="1960860">it</span> <span m="1960960">is</span> <span m="1961180">in</span> <span m="1961320">roulette,</span> <span m="1963960">then</span> <span m="1964260">this</span> <span m="1965060">is</span> <span m="1965300">a</span> <span m="1965390">number</span> <span m="1965710">bigger</span> <span m="1965970">than</span> <span m="1966150">1.</span> <span m="1967720">That</span> <span m="1967940">means</span> <span m="1968310">that</span> <span m="1969620">1</span> <span m="1969910">minus</span> <span m="1970290">p</span> <span m="1971000">over</span> <span m="1971300">p</span> <span m="1972250">is</span> <span m="1972420">bigger</span> <span m="1972670">than</span> <span m="1972830">1.</span> </p>
<p><span m="1976340">So</span> <span m="1976490">this</span> <span m="1976690">is</span> <span m="1976910">bigger</span> <span m="1977300">than</span> <span m="1977490">1.</span> <span m="1977990">This</span> <span m="1978180">is</span> <span m="1978270">bigger</span> <span m="1978470">than</span> <span m="1978630">1.</span> <span m="1978990">T</span> <span m="1980390">is</span> <span m="1980730">the</span> <span m="1980860">upper</span> <span m="1981130">limit.</span> <span m="1981620">It's</span> <span m="1981770">n</span> <span m="1981940">plus</span> <span m="1982220">m.</span> </p>
<p><span m="1983410">So</span> <span m="1983520">I've</span> <span m="1983650">got</span> <span m="1983790">a</span> <span m="1983840">bigger</span> <span m="1984280">number</span> <span m="1984590">down</span> <span m="1984900">here</span> <span m="1985190">than</span> <span m="1985370">I</span> <span m="1985460">do</span> <span m="1985670">here.</span> <span m="1985970">So</span> <span m="1986380">overall,</span> <span m="1986910">it's a</span> <span m="1987020">fraction</span> <span m="1987530">less</span> <span m="1987760">than</span> <span m="1987890">1.</span> <span m="1988750">And when</span> <span m="1988810">you</span> <span m="1988940">have</span> <span m="1989110">a</span> <span m="1989180">fraction</span> <span m="1989690">less</span> <span m="1989920">than</span> <span m="1990050">1,</span> <span m="1990860">if</span> <span m="1990990">you</span> <span m="1991120">add</span> <span m="1991470">1</span> <span m="1991750">to</span> <span m="1991860">the</span> <span m="1991930">numerator</span> <span m="1992520">and</span> <span m="1992730">denominator,</span> <span m="1994330">it</span> <span m="1994470">gets</span> <span m="1994660">closer</span> <span m="1995000">to</span> <span m="1995090">1.</span> <span m="1995480">It gets</span> <span m="1995710">bigger.</span> </p>
<p><span m="1997670">So</span> <span m="1998590">this</span> <span m="1998850">is</span> <span m="2000100">upper-bounded</span> <span m="2001800">by</span> <span m="2002030">just</span> <span m="2002330">adding</span> <span m="2002630">1</span> <span m="2002880">to</span> <span m="2002990">each</span> <span m="2003220">of</span> <span m="2003270">these.</span> <span m="2003460">Its</span> <span m="2003630">upper-bounded</span> <span m="2004300">by</span> <span m="2004430">this</span> <span m="2004900">over</span> <span m="2005080">that,</span> <span m="2006330">which</span> <span m="2006580">is</span> <span m="2007440">1</span> <span m="2007650">minus</span> <span m="2008000">p</span> <span m="2008440">over</span> <span m="2008720">p</span> <span m="2010070">to</span> <span m="2010220">the</span> <span m="2011170">n</span> <span m="2011390">minus</span> <span m="2011820">T.</span> <span m="2015060">And T is</span> <span m="2015180">just</span> <span m="2015440">n</span> <span m="2015650">plus</span> <span m="2015970">m.</span> <span m="2016860">So</span> <span m="2017120">this</span> <span m="2017430">equals--</span> <span m="2019530">why</span> <span m="2019700">don't</span> <span m="2019910">I</span> <span m="2019970">turn</span> <span m="2020190">it</span> <span m="2020280">upside</span> <span m="2020720">down?</span> <span m="2020930">Make</span> <span m="2021100">it</span> <span m="2021400">p over</span> <span m="2021700">1</span> <span m="2021900">minus</span> <span m="2022220">p</span> <span m="2022370">to</span> <span m="2022430">get</span> <span m="2022560">a</span> <span m="2022610">fraction</span> <span m="2023310">that's</span> <span m="2023430">less</span> <span m="2023660">than</span> <span m="2023800">1.</span> </p>
<p><span m="2024720">T</span> <span m="2025020">minus</span> <span m="2025390">n,</span> <span m="2026530">and</span> <span m="2026640">that</span> <span m="2026890">equals</span> <span m="2027790">p</span> <span m="2028260">over</span> <span m="2028530">1</span> <span m="2028770">minus</span> <span m="2029140">p</span> <span m="2030080">to</span> <span m="2030530">the</span> <span m="2030720">m.</span> <span m="2034240">And</span> <span m="2034350">this</span> <span m="2034490">is</span> <span m="2034590">how</span> <span m="2034700">much</span> <span m="2034890">you're</span> <span m="2034970">trying</span> <span m="2035230">to</span> <span m="2035280">get</span> <span m="2035420">ahead--</span> <span m="2035970">$100</span> <span m="2037430">in</span> <span m="2037480">the</span> <span m="2037560">case</span> <span m="2037850">of</span> <span m="2037900">my</span> <span m="2038010">mother-in-law.</span> <span m="2041230">So</span> <span m="2041880">what</span> <span m="2042170">we've</span> <span m="2042370">proved--</span> <span m="2043280">let me</span> <span m="2043660">state</span> <span m="2044040">what we</span> <span m="2044260">proved</span> <span m="2044530">as</span> <span m="2044620">a</span> <span m="2044700">theorem.</span> </p>
<p></p>
<p><span m="2064929">So</span> <span m="2065060">we</span> <span m="2065190">proved</span> <span m="2065620">that</span> <span m="2070540">if</span> <span m="2070730">p</span> <span m="2071000">is</span> <span m="2071159">less</span> <span m="2071400">than</span> <span m="2071540">1/2--</span> <span m="2071830">if</span> <span m="2072090">you're</span> <span m="2072610">more</span> <span m="2072719">likely</span> <span m="2073110">to</span> <span m="2073280">lose</span> <span m="2073590">a</span> <span m="2073650">bet</span> <span m="2073889">than</span> <span m="2074040">win</span> <span m="2074219">it--</span> <span m="2076909">then</span> <span m="2078130">the</span> <span m="2078239">probability</span> <span m="2080420">that</span> <span m="2080540">you</span> <span m="2080699">win</span> <span m="2081980">m</span> <span m="2082210">dollars</span> <span m="2084440">before</span> <span m="2085050">you</span> <span m="2085210">lose</span> <span m="2085580">n</span> <span m="2088310">dollars</span> <span m="2090290">is</span> <span m="2090520">at</span> <span m="2090610">most</span> <span m="2090880">p</span> <span m="2091150">over</span> <span m="2091949">1</span> <span m="2092190">minus</span> <span m="2092560">p</span> <span m="2093520">to</span> <span m="2093719">the</span> <span m="2093889">m.</span> <span m="2095510">That's</span> <span m="2095739">what</span> <span m="2095820">we</span> <span m="2095940">just</span> <span m="2096150">proved.</span> <span m="2098590">And</span> <span m="2098730">so</span> <span m="2098820">now</span> <span m="2099040">you</span> <span m="2099190">can</span> <span m="2099340">plug in</span> <span m="2099720">values--</span> <span m="2100230">for</span> <span m="2100330">example,</span> <span m="2100770">for</span> <span m="2101050">roulette.</span> </p>
<p><span m="2105260">p</span> <span m="2105670">equals</span> <span m="2106080">9/19,</span> <span m="2109160">which</span> <span m="2109400">means</span> <span m="2109770">that</span> <span m="2110960">p</span> <span m="2111460">over</span> <span m="2111700">1</span> <span m="2111960">minus</span> <span m="2112300">p--</span> <span m="2114170">that's</span> <span m="2114400">going</span> <span m="2114490">to</span> <span m="2114530">be</span> <span m="2114620">9/19</span> <span m="2117230">over</span> <span m="2117480">10/19,</span> <span m="2119750">which</span> <span m="2119820">is</span> <span m="2119950">just</span> <span m="2120550">9/10.</span> <span m="2124740">And</span> <span m="2125110">if m--</span> <span m="2126170">the</span> <span m="2126260">amount</span> <span m="2126480">you</span> <span m="2126570">want</span> <span m="2126720">to</span> <span m="2126790">win--</span> <span m="2127000">is</span> <span m="2127100">$100,</span> <span m="2130010">and</span> <span m="2130230">n</span> <span m="2130890">is</span> <span m="2131080">$1,000--</span> <span m="2132200">that's</span> <span m="2132310">what</span> <span m="2132480">you</span> <span m="2132570">start</span> <span m="2132890">with</span> <span m="2133165">and</span> <span m="2133440">you're</span> <span m="2133620">willing</span> <span m="2133870">to</span> <span m="2133990">lose--</span> <span m="2136270">well,</span> <span m="2136420">the</span> <span m="2136510">probability</span> <span m="2137180">you</span> <span m="2137290">win--</span> <span m="2137690">you</span> <span m="2137780">go</span> <span m="2137950">home</span> <span m="2138160">happy--</span> <span m="2142410">W</span> <span m="2142640">star</span> <span m="2143090">you</span> <span m="2143270">win</span> <span m="2143490">$100--</span> <span m="2146570">is</span> <span m="2146810">less</span> <span m="2147090">than</span> <span m="2147230">or</span> <span m="2147320">equal</span> <span m="2147570">to</span> <span m="2147690">9/10</span> <span m="2150310">raised</span> <span m="2150780">to</span> <span m="2150910">the</span> <span m="2151070">m,</span> <span m="2151300">which</span> <span m="2151450">is</span> <span m="2151530">100.</span> <span m="2153800">So</span> <span m="2153880">it's</span> <span m="2154000">9/10</span> <span m="2154550">of</span> <span m="2154610">100,</span> <span m="2155580">and</span> <span m="2155670">that</span> <span m="2155990">turns</span> <span m="2156280">out</span> <span m="2156580">to</span> <span m="2156670">be</span> <span m="2156990">less</span> <span m="2157260">than</span> <span m="2157410">1</span> <span m="2157610">in</span> <span m="2157730">37,648,</span> <span m="2162780">which</span> <span m="2162970">is</span> <span m="2163060">where</span> <span m="2163260">that</span> <span m="2163440">answer</span> <span m="2163740">came</span> <span m="2163990">from.</span> </p>
<p><span m="2166280">Now</span> <span m="2166320">you</span> <span m="2166440">can</span> <span m="2166580">see</span> <span m="2166710">why</span> <span m="2166820">my</span> <span m="2166940">mother-in-law</span> <span m="2167800">may</span> <span m="2167950">have</span> <span m="2168060">got</span> <span m="2168240">lost</span> <span m="2168520">somewhere</span> <span m="2168940">here</span> <span m="2169820">now</span> <span m="2170010">in</span> <span m="2170080">the</span> <span m="2170180">calculations.</span> <span m="2171640">But</span> <span m="2171810">this</span> <span m="2172010">is</span> <span m="2172190">a</span> <span m="2172270">proof</span> <span m="2172870">that the</span> <span m="2173150">chance</span> <span m="2173560">you</span> <span m="2174020">win</span> <span m="2174190">$100</span> <span m="2174760">before</span> <span m="2175020">you</span> <span m="2175100">lose</span> <span m="2175280">$1,000</span> <span m="2176710">is</span> <span m="2176860">very,</span> <span m="2177190">very</span> <span m="2177300">small.</span> <span m="2178850">Now,</span> <span m="2178890">do you</span> <span m="2179060">see</span> <span m="2179830">why</span> <span m="2180160">the</span> <span m="2180340">answer</span> <span m="2181150">is</span> <span m="2181330">no</span> <span m="2181500">better</span> <span m="2181850">than</span> <span m="2182060">if</span> <span m="2182190">you</span> <span m="2182320">came</span> <span m="2182810">with</span> <span m="2182980">$1</span> <span m="2183030">million</span> <span m="2183810">in</span> <span m="2183850">your</span> <span m="2183950">pocket?</span> </p>
<p><span m="2186310">Say</span> <span m="2186490">you</span> <span m="2186680">came</span> <span m="2186940">with</span> <span m="2187120">n</span> <span m="2187350">equals</span> <span m="2187630">$1</span> <span m="2187690">million.</span> <span m="2190390">Why</span> <span m="2190640">is</span> <span m="2190770">the</span> <span m="2190910">answer</span> <span m="2191940">not</span> <span m="2192130">changing?</span> <span m="2195690">Yeah.</span> </p>
<p><span m="2197020">AUDIENCE: Once</span> <span m="2197460">you</span> <span m="2197900">lose,</span> <span m="2198340">say,</span> <span m="2198780">$1,000,</span> <span m="2199930">you're</span> <span m="2200130">already</span> <span m="2200490">in</span> <span m="2200860">a</span> <span m="2200960">really</span> <span m="2201210">deep</span> <span m="2201380">hole.</span> </p>
<p><span m="2202650">PROFESSOR: That's</span> <span m="2202900">the</span> <span m="2203020">intuition.</span> <span m="2204130">That's</span> <span m="2204340">right.</span> <span m="2204490">We're</span> <span m="2204550">going</span> <span m="2204720">to</span> <span m="2204760">get</span> <span m="2204940">to</span> <span m="2205020">that</span> <span m="2205260">in</span> <span m="2205360">a</span> <span m="2205460">minute.</span> <span m="2206560">I</span> <span m="2206680">want</span> <span m="2206860">to</span> <span m="2206920">know</span> <span m="2207310">from</span> <span m="2207540">the</span> <span m="2207630">formula,</span> <span m="2209250">why</span> <span m="2209590">is</span> <span m="2209740">it</span> <span m="2209900">no</span> <span m="2210130">difference</span> <span m="2210700">if</span> <span m="2210820">I</span> <span m="2210900">come</span> <span m="2211240">with</span> <span m="2211480">$1,000</span> <span m="2212300">versus</span> <span m="2212720">$1</span> <span m="2212790">million?</span> <span m="2214280">Yeah.</span> </p>
<p><span m="2215248">AUDIENCE: The formula</span> <span m="2215696">doesn't have</span> <span m="2216144">n.</span> </p>
<p><span m="2216592">PROFESSOR: Yeah,</span> <span m="2217040">the</span> <span m="2217170">formula</span> <span m="2217650">has</span> <span m="2217870">nothing</span> <span m="2218310">to</span> <span m="2218390">do</span> <span m="2218590">with</span> <span m="2218740">n.</span> <span m="2220180">You</span> <span m="2220320">could</span> <span m="2220500">come</span> <span m="2220730">with</span> <span m="2220830">$100</span> <span m="2221320">trillion</span> <span m="2222190">in</span> <span m="2222260">your</span> <span m="2222370">wallet,</span> <span m="2224130">and</span> <span m="2224290">it</span> <span m="2224400">doesn't</span> <span m="2224660">improve</span> <span m="2225120">this</span> <span m="2225330">bound.</span> <span m="2226710">This</span> <span m="2226940">bound</span> <span m="2227280">only</span> <span m="2227570">depends</span> <span m="2228060">on</span> <span m="2228160">what</span> <span m="2228330">you're</span> <span m="2228440">trying</span> <span m="2228790">to</span> <span m="2228990">win,</span> <span m="2230260">not</span> <span m="2230520">on</span> <span m="2230610">how</span> <span m="2230780">much</span> <span m="2230970">you</span> <span m="2231050">came</span> <span m="2231340">with.</span> <span m="2231530">So</span> <span m="2231640">no</span> <span m="2231720">matter</span> <span m="2231970">how</span> <span m="2232280">much</span> <span m="2232500">you</span> <span m="2232590">come</span> <span m="2232860">with,</span> <span m="2233890">the</span> <span m="2234000">chance</span> <span m="2234300">you</span> <span m="2234410">win</span> <span m="2234590">$100</span> <span m="2235080">before</span> <span m="2235350">you</span> <span m="2235480">lose</span> <span m="2235730">everything</span> <span m="2237250">is</span> <span m="2237460">at</span> <span m="2237540">most</span> <span m="2237880">1</span> <span m="2238303">in 37,000.</span> </p>
<p><span m="2240860">Now,</span> <span m="2241350">we</span> <span m="2241520">can</span> <span m="2241630">plug in</span> <span m="2241950">some</span> <span m="2242100">other</span> <span m="2242260">values</span> <span m="2242650">just</span> <span m="2242870">for</span> <span m="2242980">fun--</span> <span m="2245870">different</span> <span m="2246170">values</span> <span m="2246530">of</span> <span m="2246600">m.</span> <span m="2252380">If</span> <span m="2252450">you</span> <span m="2252560">thought</span> <span m="2252920">1</span> <span m="2253160">in</span> <span m="2253260">37,000</span> <span m="2254230">was</span> <span m="2254390">unlikely,</span> <span m="2256350">the</span> <span m="2256470">chance</span> <span m="2256860">of</span> <span m="2256990">winning</span> <span m="2257290">$1,000,</span> <span m="2260776">or</span> <span m="2261280">1,000</span> <span m="2261810">bets</span> <span m="2262080">worth</span> <span m="2264380">before</span> <span m="2264760">you're</span> <span m="2264870">broke--</span> <span m="2266230">that's</span> <span m="2266520">less than</span> <span m="2266880">9/10</span> <span m="2268170">to</span> <span m="2268600">the</span> <span m="2268740">1,000.</span> <span m="2270870">That's</span> <span m="2271200">less</span> <span m="2271480">than</span> <span m="2273350">2</span> <span m="2273680">times</span> <span m="2274040">10</span> <span m="2274380">to</span> <span m="2274480">the</span> <span m="2274530">minus</span> <span m="2275020">46--</span> <span m="2277090">really,</span> <span m="2277450">really,</span> <span m="2278120">really</span> <span m="2278440">unlikely.</span> <span m="2279760">Even</span> <span m="2280150">winning</span> <span m="2280690">$10</span> <span m="2283830">is</span> <span m="2284040">not</span> <span m="2284410">likely.</span> </p>
<p><span m="2286400">Just</span> <span m="2286580">plug in</span> <span m="2286880">the</span> <span m="2287240">numbers.</span> <span m="2287650">The probability</span> <span m="2288110">you</span> <span m="2288220">win</span> <span m="2289320">$10</span> <span m="2290680">betting</span> <span m="2291040">$1</span> <span m="2291470">at</span> <span m="2291530">a</span> <span m="2291610">time</span> <span m="2293790">is</span> <span m="2293960">less</span> <span m="2294230">than</span> <span m="2294370">9/10</span> <span m="2295080">to</span> <span m="2295180">the</span> <span m="2295280">10th</span> <span m="2296430">power.</span> <span m="2298470">That's</span> <span m="2298760">less</span> <span m="2299040">than</span> <span m="2300150">0.35.</span> </p>
<p><span m="2304150">You can</span> <span m="2304450">come</span> <span m="2304700">to</span> <span m="2304740">the</span> <span m="2304850">casino</span> <span m="2305370">with</span> <span m="2305610">$10</span> <span m="2306090">million,</span> <span m="2307785">bet</span> <span m="2308200">$1</span> <span m="2308640">at</span> <span m="2308700">a</span> <span m="2308780">time,</span> <span m="2310710">and</span> <span m="2310850">you</span> <span m="2310960">quit</span> <span m="2311320">if</span> <span m="2311430">you</span> <span m="2311510">just</span> <span m="2311850">get</span> <span m="2311960">up</span> <span m="2312110">10</span> <span m="2312390">bets--</span> <span m="2313230">get</span> <span m="2313350">up</span> <span m="2313490">$10.</span> <span m="2314390">The</span> <span m="2314500">chance</span> <span m="2315020">you</span> <span m="2315120">get</span> <span m="2315270">up</span> <span m="2315400">$10</span> <span m="2315900">before</span> <span m="2316160">you</span> <span m="2316290">lose</span> <span m="2316570">$10</span> <span m="2316790">million</span> <span m="2317890">is</span> <span m="2318360">about</span> <span m="2318580">1</span> <span m="2318740">in</span> <span m="2318840">3</span> <span m="2320010">you're</span> <span m="2320190">twice</span> <span m="2320680">as</span> <span m="2320810">likely</span> <span m="2321200">to</span> <span m="2321290">lose</span> <span m="2321570">$10</span> <span m="2321800">million</span> <span m="2322100">as</span> <span m="2322620">you</span> <span m="2322790">are</span> <span m="2322940">to</span> <span m="2323010">win</span> <span m="2323250">10.</span> <span m="2324400">That</span> <span m="2324490">just</span> <span m="2324700">seems</span> <span m="2324860">weird,</span> <span m="2325670">right?</span> </p>
<p><span m="2326780">Because</span> <span m="2326860">it's</span> <span m="2327010">almost</span> <span m="2327500">a</span> <span m="2327550">fair</span> <span m="2327880">game.</span> <span m="2329250">It's</span> <span m="2329450">almost</span> <span m="2330700">50-50.</span> <span m="2331390">Any</span> <span m="2331560">questions</span> <span m="2332140">about</span> <span m="2332510">the</span> <span m="2332710">analysis?</span> </p>
<p><span m="2335860">Yes,</span> <span m="2336310">I</span> <span m="2336650">find</span> <span m="2336950">that</span> <span m="2337300">shocking.</span> <span m="2339710">Just</span> <span m="2340010">the</span> <span m="2340100">intuition</span> <span m="2340600">would</span> <span m="2340720">seem</span> <span m="2341010">say</span> <span m="2342040">otherwise.</span> <span m="2342580">So</span> <span m="2343050">I</span> <span m="2343120">guess</span> <span m="2343290">there's</span> <span m="2343410">a</span> <span m="2343470">moral</span> <span m="2343800">here.</span> </p>
<p><span m="2343990">If</span> <span m="2344160">you're</span> <span m="2344260">going</span> <span m="2344390">to</span> <span m="2344470">gamble,</span> <span m="2345530">learn</span> <span m="2345750">how</span> <span m="2345880">to</span> <span m="2345970">count</span> <span m="2346220">cards</span> <span m="2346660">in</span> <span m="2346740">blackjack,</span> <span m="2347840">or</span> <span m="2347990">some</span> <span m="2348320">game</span> <span m="2348850">where</span> <span m="2349050">you</span> <span m="2349190">can</span> <span m="2349370">make</span> <span m="2349670">it</span> <span m="2349890">even.</span> <span m="2351060">Because</span> <span m="2351260">even</span> <span m="2351500">in</span> <span m="2351560">a</span> <span m="2351610">game</span> <span m="2351910">where</span> <span m="2352020">it's</span> <span m="2352130">pretty</span> <span m="2352310">close,</span> <span m="2352680">you're</span> <span m="2352850">doomed.</span> <span m="2354580">You're</span> <span m="2354710">just</span> <span m="2354890">never</span> <span m="2355110">going</span> <span m="2355230">to</span> <span m="2355290">go</span> <span m="2355830">home</span> <span m="2356090">happy.</span> </p>
<p><span m="2358710">Now,</span> <span m="2359540">if</span> <span m="2359760">you</span> <span m="2359890">could</span> <span m="2360020">have</span> <span m="2360180">a</span> <span m="2360250">fair</span> <span m="2360660">game,</span> <span m="2361900">the</span> <span m="2362010">world</span> <span m="2362520">changes--</span> <span m="2363410">much</span> <span m="2363750">better</span> <span m="2363990">circumstance.</span> <span m="2364780">So</span> <span m="2365260">actually,</span> <span m="2365570">let's</span> <span m="2365820">do</span> <span m="2365900">the</span> <span m="2366050">same</span> <span m="2366470">analysis</span> <span m="2367740">for</span> <span m="2367930">a</span> <span m="2367990">fair</span> <span m="2368310">game,</span> <span m="2368800">because</span> <span m="2368990">that's</span> <span m="2369220">where</span> <span m="2369370">our</span> <span m="2369490">intuition</span> <span m="2370130">really</span> <span m="2370380">comes</span> <span m="2370680">from.</span> <span m="2371680">Because</span> <span m="2371840">we're</span> <span m="2371960">thinking</span> <span m="2372350">of</span> <span m="2372420">this</span> <span m="2372600">game</span> <span m="2372810">as</span> <span m="2372920">almost</span> <span m="2373430">fair.</span> </p>
<p><span m="2374550">And</span> <span m="2374770">in a</span> <span m="2374830">fair</span> <span m="2375180">game,</span> <span m="2375960">the</span> <span m="2376110">answer's</span> <span m="2376480">going</span> <span m="2376600">to</span> <span m="2376670">be</span> <span m="2376860">very</span> <span m="2377330">different.</span> <span m="2378700">And</span> <span m="2378860">it</span> <span m="2379100">all</span> <span m="2379450">goes</span> <span m="2379720">back</span> <span m="2380020">to</span> <span m="2380130">the</span> <span m="2380230">recurrence</span> <span m="2380840">and</span> <span m="2380920">the</span> <span m="2381030">roots</span> <span m="2381320">of</span> <span m="2381370">the</span> <span m="2381440">characteristic</span> <span m="2382030">equation.</span> <span m="2383490">Because</span> <span m="2383800">in</span> <span m="2383900">a</span> <span m="2383950">fair</span> <span m="2384320">game,</span> <span m="2388920">p</span> <span m="2389180">is</span> <span m="2389310">1/2.</span> </p>
<p><span m="2393490">And</span> <span m="2393710">then</span> <span m="2394360">you</span> <span m="2394410">have</span> <span m="2394550">a</span> <span m="2394630">double</span> <span m="2394920">root.</span> <span m="2396290">1</span> <span m="2396450">minus</span> <span m="2396770">1/2</span> <span m="2397210">over</span> <span m="2397440">1/2</span> <span m="2397960">equals</span> <span m="2398320">1,</span> <span m="2399750">and</span> <span m="2399960">that</span> <span m="2400220">means</span> <span m="2401020">a</span> <span m="2401150">double</span> <span m="2401580">root</span> <span m="2404970">at</span> <span m="2405170">1.</span> <span m="2407570">And</span> <span m="2407700">that</span> <span m="2408040">changes</span> <span m="2408910">everything.</span> </p>
<p><span m="2410390">So</span> <span m="2410900">let's</span> <span m="2411170">go</span> <span m="2411330">through</span> <span m="2411560">now</span> <span m="2411830">and</span> <span m="2411960">do</span> <span m="2412160">all</span> <span m="2412330">this</span> <span m="2412520">analysis</span> <span m="2416120">in</span> <span m="2416290">the</span> <span m="2416390">case</span> <span m="2416720">of</span> <span m="2416820">a</span> <span m="2416870">fair</span> <span m="2417150">game.</span> <span m="2420140">And</span> <span m="2420270">this</span> <span m="2420430">will</span> <span m="2420500">give</span> <span m="2420650">us</span> <span m="2420770">practice</span> <span m="2421260">with</span> <span m="2421630">double</span> <span m="2421880">roots</span> <span m="2422180">and</span> <span m="2422360">recurrences.</span> <span m="2423630">Because</span> <span m="2424080">as you</span> <span m="2424180">see</span> <span m="2424390">now,</span> <span m="2424990">it</span> <span m="2425140">does</span> <span m="2425440">happen.</span> </p>
<p><span m="2430930">Let's</span> <span m="2431170">figure</span> <span m="2431430">out</span> <span m="2431520">the</span> <span m="2431600">chance</span> <span m="2432000">that</span> <span m="2432090">we</span> <span m="2432260">go</span> <span m="2432480">home</span> <span m="2432942">a</span> <span m="2433404">winner.</span> <span m="2438490">OK,</span> <span m="2438780">so</span> <span m="2439970">let's</span> <span m="2440180">see.</span> <span m="2440360">In</span> <span m="2440470">this</span> <span m="2440690">case,</span> <span m="2442400">we</span> <span m="2442510">know</span> <span m="2442680">the</span> <span m="2442810">roots.</span> <span m="2444766">Can anybody</span> <span m="2445230">tell</span> <span m="2445420">me</span> <span m="2445530">what</span> <span m="2445740">formula</span> <span m="2446430">we're</span> <span m="2446580">going</span> <span m="2446710">to</span> <span m="2446760">use</span> <span m="2447230">for</span> <span m="2447650">the</span> <span m="2447790">solution?</span> <span m="2449870">Got</span> <span m="2449960">a</span> <span m="2450030">double</span> <span m="2450430">root</span> <span m="2451100">at</span> <span m="2451310">1.</span> <span m="2452230">So</span> <span m="2452340">there's</span> <span m="2452520">going</span> <span m="2452650">to</span> <span m="2452710">be</span> <span m="2452870">a</span> <span m="2452940">1</span> <span m="2453250">to</span> <span m="2453320">the</span> <span m="2453480">n</span> <span m="2453670">here.</span> <span m="2456750">I</span> <span m="2456860">don't</span> <span m="2457060">just</span> <span m="2457250">put</span> <span m="2457420">a</span> <span m="2457530">constant</span> <span m="2458000">A</span> <span m="2458370">in front.</span> <span m="2458590">What</span> <span m="2458720">do</span> <span m="2458810">I</span> <span m="2458870">do</span> <span m="2459000">with</span> <span m="2459160">a</span> <span m="2459220">double</span> <span m="2459580">root?</span> </p>
<p><span m="2460850">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2461660">AUDIENCE: A n.</span> </p>
<p><span m="2462260">PROFESSOR: What</span> <span m="2462420">is</span> <span m="2462550">it?</span> </p>
<p><span m="2463430">AUDIENCE: A n.</span> </p>
<p><span m="2464230">PROFESSOR: A n--</span> <span m="2465720">not</span> <span m="2465990">quite</span> <span m="2466130">A n.</span> <span m="2466560">You</span> <span m="2466630">got</span> <span m="2466940">an A n</span> <span m="2467230">here.</span> </p>
<p><span m="2468200">AUDIENCE: Plus B.</span> </p>
<p><span m="2469080">PROFESSOR: Plus</span> <span m="2469440">B--</span> <span m="2470720">that's</span> <span m="2470980">what</span> <span m="2471090">you</span> <span m="2471190">do</span> <span m="2471370">for</span> <span m="2471500">a</span> <span m="2471540">double</span> <span m="2471830">root,</span> <span m="2473280">because</span> <span m="2473440">you</span> <span m="2473730">make</span> <span m="2473960">a</span> <span m="2474030">first</span> <span m="2474320">degree</span> <span m="2474630">polynomial</span> <span m="2475260">in</span> <span m="2475330">n</span> <span m="2475880">here.</span> <span m="2477340">So</span> <span m="2477460">we</span> <span m="2477560">plug</span> <span m="2477810">that</span> <span m="2478030">in.</span> <span m="2478870">The</span> <span m="2478980">root's at</span> <span m="2479280">1,</span> <span m="2479530">so</span> <span m="2479640">it's</span> <span m="2479750">real</span> <span m="2480040">easy.</span> </p>
<p><span m="2480240">The</span> <span m="2480370">solution's</span> <span m="2481030">really</span> <span m="2481380">easy</span> <span m="2481690">now.</span> <span m="2481910">No</span> <span m="2482120">messy</span> <span m="2482650">powers</span> <span m="2483020">or</span> <span m="2483230">anything.</span> <span m="2483340">It's</span> <span m="2483470">just</span> <span m="2483680">A</span> <span m="2483830">n</span> <span m="2484070">plus</span> <span m="2484220">B.</span> <span m="2485900">And</span> <span m="2486100">I</span> <span m="2486150">can</span> <span m="2486310">figure</span> <span m="2486720">out</span> <span m="2487020">A and</span> <span m="2487370">B</span> <span m="2488610">from</span> <span m="2488830">the</span> <span m="2488920">boundary</span> <span m="2489330">conditions.</span> </p>
<p><span m="2496550">All</span> <span m="2496730">right,</span> <span m="2497670">X0</span> <span m="2498330">is</span> <span m="2498490">0.</span> <span m="2500590">X</span> <span m="2501040">0</span> <span m="2501430">is</span> <span m="2501580">just</span> <span m="2501910">B,</span> <span m="2503990">because</span> <span m="2504200">it's</span> <span m="2504610">A</span> <span m="2504770">times</span> <span m="2505110">0</span> <span m="2505270">goes</span> <span m="2505480">away.</span> <span m="2506280">And</span> <span m="2506410">that</span> <span m="2506580">means</span> <span m="2506830">that</span> <span m="2506970">B</span> <span m="2507360">equals</span> <span m="2507640">0.</span> <span m="2508120">This</span> <span m="2508300">is</span> <span m="2508420">getting</span> <span m="2508690">really</span> <span m="2508940">simple.</span> </p>
<p><span m="2510960">1</span> <span m="2511460">is</span> <span m="2511700">X T.</span> <span m="2514070">And</span> <span m="2514200">that's</span> <span m="2514570">A</span> <span m="2514770">plus</span> <span m="2515130">B,</span> <span m="2515380">but</span> <span m="2515520">B</span> <span m="2515740">was</span> <span m="2515920">0.</span> <span m="2516850">So</span> <span m="2517010">that's</span> <span m="2517250">A</span> <span m="2518240">times</span> <span m="2518430">1</span> <span m="2518700">plus</span> <span m="2520000">B.</span> <span m="2520900">That's</span> <span m="2521110">just</span> <span m="2521270">A.</span> </p>
<p><span m="2523620">It</span> <span m="2523760">means</span> <span m="2524140">A</span> <span m="2525130">equals</span> <span m="2530380">A n.</span> <span m="2531600">Good,</span> <span m="2532400">n's</span> <span m="2532680">not</span> <span m="2532890">1.</span> <span m="2533270">N's</span> <span m="2533600">T.</span> <span m="2534370">So</span> <span m="2534490">it's</span> <span m="2534610">A</span> <span m="2534750">T</span> <span m="2534980">plus</span> <span m="2535240">B.</span> </p>
<p><span m="2535410">This</span> <span m="2535600">is</span> <span m="2535720">A</span> <span m="2535860">T</span> <span m="2536310">here.</span> <span m="2537550">So</span> <span m="2537820">A</span> <span m="2538030">T</span> <span m="2538450">equals</span> <span m="2538750">1.</span> <span m="2539280">That</span> <span m="2539430">means</span> <span m="2539750">A</span> <span m="2540060">is</span> <span m="2540240">1</span> <span m="2540480">over</span> <span m="2540670">T.</span> </p>
<p><span m="2543780">All</span> <span m="2543850">right,</span> <span m="2544020">that</span> <span m="2544360">means</span> <span m="2544810">that</span> <span m="2546990">X</span> <span m="2547300">n</span> <span m="2548580">is</span> <span m="2550160">n</span> <span m="2550770">over</span> <span m="2551070">T.</span> <span m="2551962">And</span> <span m="2552408">T</span> <span m="2552854">is</span> <span m="2553300">the</span> <span m="2553430">total.</span> <span m="2553820">The</span> <span m="2554070">top</span> <span m="2554330">limit</span> <span m="2554700">is</span> <span m="2554910">n</span> <span m="2555140">plus</span> <span m="2555390">m,</span> <span m="2555770">because you</span> <span m="2555890">quit</span> <span m="2556140">if</span> <span m="2556240">you</span> <span m="2556310">get</span> <span m="2556460">ahead</span> <span m="2556750">m</span> <span m="2556970">dollars.</span> </p>
<p><span m="2558090">This</span> <span m="2558320">is</span> <span m="2558460">just</span> <span m="2558770">now</span> <span m="2559050">n</span> <span m="2559630">over</span> <span m="2560090">n</span> <span m="2560660">plus</span> <span m="2561110">m.</span> <span m="2564030">All</span> <span m="2564190">right,</span> <span m="2564510">so</span> <span m="2566340">let's</span> <span m="2566520">write</span> <span m="2566780">that</span> <span m="2567010">down.</span> <span m="2567880">It's a</span> <span m="2568040">theorem.</span> </p>
<p><span m="2573450">If</span> <span m="2573700">p</span> <span m="2573920">is</span> <span m="2574050">1/2,</span> <span m="2574610">i.e.,</span> <span m="2575330">you</span> <span m="2575530">have</span> <span m="2575660">a</span> <span m="2575720">fair</span> <span m="2575980">game,</span> <span m="2578010">then</span> <span m="2578920">the</span> <span m="2579050">probability</span> <span m="2580770">you</span> <span m="2580960">win</span> <span m="2582510">m</span> <span m="2582720">dollars</span> <span m="2584580">before</span> <span m="2585030">you</span> <span m="2585200">lose</span> <span m="2588190">n</span> <span m="2588490">dollars</span> <span m="2590650">is</span> <span m="2590830">just</span> <span m="2591160">n</span> <span m="2591850">over</span> <span m="2592210">n</span> <span m="2592370">plus</span> <span m="2592660">m.</span> <span m="2595270">And</span> <span m="2595510">this</span> <span m="2595940">might</span> <span m="2596290">fit</span> <span m="2596500">the</span> <span m="2596630">intuition</span> <span m="2597220">better.</span> <span m="2600700">So</span> <span m="2601160">for</span> <span m="2601290">the</span> <span m="2601390">mother-in-law</span> <span m="2601980">strategy,</span> <span m="2605810">if</span> <span m="2606070">m is</span> <span m="2606480">100,</span> <span m="2608810">and</span> <span m="2609280">n</span> <span m="2609520">is</span> <span m="2609650">1,000,</span> <span m="2613440">what's</span> <span m="2613710">the</span> <span m="2613810">probability</span> <span m="2614390">you win--</span> <span m="2615100">you</span> <span m="2615240">go</span> <span m="2615370">home</span> <span m="2615500">a</span> <span m="2615980">winner?</span> </p>
<p><span m="2622230">Yeah,</span> <span m="2622530">1,000</span> <span m="2624140">over</span> <span m="2624690">1,000</span> <span m="2626870">plus</span> <span m="2627180">100.</span> <span m="2628904">1,000</span> <span m="2629770">over</span> <span m="2629900">1,000</span> <span m="2630590">is</span> <span m="2631700">10</span> <span m="2631940">over</span> <span m="2632150">11.</span> <span m="2634490">So</span> <span m="2634690">she</span> <span m="2634870">does</span> <span m="2635160">go</span> <span m="2635350">home</span> <span m="2635610">happy</span> <span m="2635930">most</span> <span m="2636220">of</span> <span m="2636320">the</span> <span m="2636360">time--</span> <span m="2636970">10</span> <span m="2637310">out</span> <span m="2637430">of</span> <span m="2637500">11</span> <span m="2637850">nights--</span> <span m="2639160">if</span> <span m="2639300">she's</span> <span m="2639480">playing</span> <span m="2639780">a</span> <span m="2639830">fair</span> <span m="2640150">game.</span> </p>
<p><span m="2642970">Any</span> <span m="2643180">questions</span> <span m="2645090">about</span> <span m="2645320">that?</span> <span m="2646920">So</span> <span m="2647100">the</span> <span m="2647430">trouble</span> <span m="2647820">we</span> <span m="2647940">get</span> <span m="2648120">into</span> <span m="2648620">here</span> <span m="2650390">is</span> <span m="2650650">that</span> <span m="2652810">the</span> <span m="2652960">fair</span> <span m="2653360">game</span> <span m="2653630">results</span> <span m="2654420">match</span> <span m="2654750">our</span> <span m="2654870">intuition.</span> <span m="2656890">You</span> <span m="2656980">know</span> <span m="2657120">if</span> <span m="2657190">you</span> <span m="2658630">have</span> <span m="2658740">10 times</span> <span m="2659060">as</span> <span m="2659250">much</span> <span m="2659450">money</span> <span m="2659780">in</span> <span m="2659870">a</span> <span m="2659920">fair</span> <span m="2660190">game,</span> <span m="2660520">you'd</span> <span m="2660620">expect</span> <span m="2661650">to</span> <span m="2661900">go</span> <span m="2662070">home</span> <span m="2662280">happy</span> <span m="2662610">10</span> <span m="2662830">out of</span> <span m="2662940">11</span> <span m="2663370">nights.</span> </p>
<p><span m="2663660">That</span> <span m="2663830">makes</span> <span m="2664040">a</span> <span m="2664100">lot</span> <span m="2664370">of</span> <span m="2664430">sense.</span> <span m="2664790">You</span> <span m="2665690">go</span> <span m="2665800">home</span> <span m="2665980">happy</span> <span m="2666260">10,</span> <span m="2666670">and then</span> <span m="2666770">you</span> <span m="2666890">lose</span> <span m="2668100">the</span> <span m="2668200">11th.</span> <span m="2668660">That's</span> <span m="2668840">a</span> <span m="2668910">10</span> <span m="2669140">to</span> <span m="2669220">1</span> <span m="2669390">ratio,</span> <span m="2669880">which</span> <span m="2669960">is</span> <span m="2670040">the</span> <span m="2670640">money you</span> <span m="2670940">brought</span> <span m="2671200">into</span> <span m="2671390">the</span> <span m="2671480">game.</span> </p>
<p><span m="2673001">The</span> <span m="2673380">trouble</span> <span m="2673680">we</span> <span m="2673780">get</span> <span m="2673910">into</span> <span m="2674710">is,</span> <span m="2675260">the</span> <span m="2675370">fair</span> <span m="2675730">game</span> <span m="2676160">is</span> <span m="2676340">very</span> <span m="2676760">close</span> <span m="2677110">to</span> <span m="2677200">the</span> <span m="2677290">real</span> <span m="2677520">game.</span> <span m="2678790">Instead</span> <span m="2678930">of</span> <span m="2679030">50-50,</span> <span m="2679770">it's</span> <span m="2679960">47-53.</span> <span m="2682120">And</span> <span m="2682350">so</span> <span m="2682520">our</span> <span m="2682700">intuition</span> <span m="2683210">says</span> <span m="2683590">the</span> <span m="2683700">results--</span> <span m="2684310">the</span> <span m="2684400">probability</span> <span m="2684860">of</span> <span m="2684920">going</span> <span m="2685120">home</span> <span m="2685320">happy</span> <span m="2685730">in</span> <span m="2685820">a</span> <span m="2685870">fair</span> <span m="2686150">game--</span> <span m="2686780">should</span> <span m="2687020">be</span> <span m="2687130">close</span> <span m="2688230">to</span> <span m="2688380">the</span> <span m="2688450">probability</span> <span m="2688910">of going</span> <span m="2689200">home</span> <span m="2689400">happy</span> <span m="2689760">in</span> <span m="2689830">the</span> <span m="2689920">real</span> <span m="2690110">game.</span> </p>
<p><span m="2691180">And</span> <span m="2691300">that's</span> <span m="2691600">not</span> <span m="2691790">true.</span> <span m="2692340">There's</span> <span m="2692420">a</span> <span m="2692490">discontinuity</span> <span m="2693410">here</span> <span m="2694090">because</span> <span m="2694550">of</span> <span m="2694670">the</span> <span m="2694760">double</span> <span m="2695050">root.</span> <span m="2695990">And</span> <span m="2696070">the</span> <span m="2696170">character</span> <span m="2696650">completely</span> <span m="2697180">changes.</span> </p>
<p><span m="2698350">So</span> <span m="2698390">instead</span> <span m="2698700">of</span> <span m="2698800">being</span> <span m="2699010">close</span> <span m="2699370">to</span> <span m="2699500">10</span> <span m="2699770">out</span> <span m="2699890">of</span> <span m="2699970">11,</span> <span m="2701380">you're</span> <span m="2701510">down</span> <span m="2701800">there</span> <span m="2702010">at 1</span> <span m="2702260">in</span> <span m="2702340">37,000--</span> <span m="2704140">completely</span> <span m="2704710">different</span> <span m="2705250">behavior.</span> <span m="2707640">OK,</span> <span m="2709610">any</span> <span m="2709770">questions?</span> <span m="2712580">All right,</span> <span m="2712750">so let</span> <span m="2712910">me</span> <span m="2713070">give</span> <span m="2713230">you</span> <span m="2713290">an--</span> <span m="2713370">yeah.</span> </p>
<p><span m="2714398">AUDIENCE: So</span> <span m="2714846">what happens</span> <span m="2715294">if you make</span> <span m="2715742">n</span> <span m="2716190">1,</span> <span m="2717540">and</span> <span m="2717750">then</span> <span m="2718360">you</span> <span m="2718520">do</span> <span m="2718650">that</span> <span m="2718840">repeatedly?</span> </p>
<p><span m="2720520">PROFESSOR: Now,</span> <span m="2720750">if</span> <span m="2720910">I</span> <span m="2720970">did</span> <span m="2721290">n</span> <span m="2721510">equals</span> <span m="2721840">1,</span> <span m="2723280">I</span> <span m="2723450">could</span> <span m="2723600">use</span> <span m="2723800">that</span> <span m="2724020">as</span> <span m="2724090">an</span> <span m="2724170">upper</span> <span m="2724350">bound,</span> <span m="2724750">and it's</span> <span m="2724910">not</span> <span m="2725110">so</span> <span m="2725200">interesting</span> <span m="2725470">as,</span> <span m="2725680">say,</span> <span m="2725840">90%.</span> <span m="2728080">But</span> <span m="2728530">I</span> <span m="2728580">would</span> <span m="2728720">actually</span> <span m="2728990">go</span> <span m="2729130">plug</span> <span m="2729440">it</span> <span m="2729570">back</span> <span m="2729840">in</span> <span m="2729960">here.</span> <span m="2730890">So</span> <span m="2730930">this</span> <span m="2731130">would</span> <span m="2731220">be</span> <span m="2731380">n</span> <span m="2731560">plus</span> <span m="2731850">1,</span> <span m="2732670">and</span> <span m="2732810">it</span> <span m="2732850">would</span> <span m="2732950">depend</span> <span m="2733300">how</span> <span m="2733420">much</span> <span m="2733650">money</span> <span m="2733930">I</span> <span m="2733990">brought.</span> </p>
<p><span m="2734890">But</span> <span m="2735040">there is</span> <span m="2735410">a</span> <span m="2735500">pretty</span> <span m="2735760">good</span> <span m="2735920">chance</span> <span m="2736230">I</span> <span m="2736330">go</span> <span m="2736500">home</span> <span m="2736690">a</span> <span m="2736750">winner</span> <span m="2737850">for</span> <span m="2739440">m</span> <span m="2739640">equals</span> <span m="2739880">1.</span> <span m="2741230">Because</span> <span m="2741530">I've</span> <span m="2741580">got</span> <span m="2741710">a</span> <span m="2741780">pretty</span> <span m="2742110">good</span> <span m="2742270">chance</span> <span m="2742760">that I</span> <span m="2742830">either--</span> <span m="2743940">47%</span> <span m="2744520">chance</span> <span m="2745160">I</span> <span m="2745200">win</span> <span m="2745410">the</span> <span m="2745480">first</span> <span m="2745740">time.</span> <span m="2746090">Then</span> <span m="2746240">I go</span> <span m="2746350">home</span> <span m="2746620">happy.</span> </p>
<p><span m="2747752">If</span> <span m="2748120">I</span> <span m="2748230">lost</span> <span m="2748660">the</span> <span m="2748750">first</span> <span m="2749010">time,</span> <span m="2749260">now</span> <span m="2749440">I've</span> <span m="2749520">just</span> <span m="2749690">got</span> <span m="2749800">to</span> <span m="2749860">win</span> <span m="2750090">twice.</span> <span m="2750790">And</span> <span m="2750860">I</span> <span m="2750910">might</span> <span m="2751150">win</span> <span m="2751330">twice</span> <span m="2751670">in</span> <span m="2751750">a</span> <span m="2751810">row.</span> <span m="2752720">That'll</span> <span m="2752960">happen</span> <span m="2753280">about</span> <span m="2753500">20%</span> <span m="2754080">of</span> <span m="2754120">the</span> <span m="2754220">time.</span> </p>
<p><span m="2755330">If</span> <span m="2755730">I</span> <span m="2755870">lose</span> <span m="2756280">that,</span> <span m="2756890">now</span> <span m="2757020">I've</span> <span m="2757120">got</span> <span m="2757240">to</span> <span m="2757280">win</span> <span m="2757450">three</span> <span m="2757920">in</span> <span m="2758020">a</span> <span m="2758090">row.</span> <span m="2758940">That'll</span> <span m="2759210">happen</span> <span m="2759580">around</span> <span m="2760140">10%</span> <span m="2760670">of</span> <span m="2760720">the</span> <span m="2760820">time.</span> <span m="2762150">So</span> <span m="2762290">I've</span> <span m="2762350">got</span> <span m="2762580">10</span> <span m="2762970">plus</span> <span m="2763250">20</span> <span m="2763760">plus</span> <span m="2764090">almost</span> <span m="2764460">50.</span> <span m="2765540">Most of</span> <span m="2765980">the</span> <span m="2766000">time,</span> <span m="2766250">I'm</span> <span m="2766340">going</span> <span m="2766450">to</span> <span m="2766490">go</span> <span m="2766680">home</span> <span m="2766860">happy</span> <span m="2767410">if</span> <span m="2767720">I</span> <span m="2767810">just</span> <span m="2768010">have</span> <span m="2768100">to</span> <span m="2768190">get</span> <span m="2768300">ahead</span> <span m="2768510">by</span> <span m="2768610">$1.</span> </p>
<p><span m="2770460">But</span> <span m="2770670">it</span> <span m="2770740">doesn't</span> <span m="2771040">take</span> <span m="2771240">much</span> <span m="2771450">more</span> <span m="2771630">than</span> <span m="2771890">one</span> <span m="2772460">before</span> <span m="2772750">you're</span> <span m="2772850">not</span> <span m="2773190">likely</span> <span m="2773460">to</span> <span m="2773520">go</span> <span m="2773640">home</span> <span m="2773840">happy.</span> <span m="2774660">Getting</span> <span m="2774880">ahead</span> <span m="2775150">10</span> <span m="2775350">is</span> <span m="2775730">not</span> <span m="2775950">going</span> <span m="2776050">to</span> <span m="2776090">happen,</span> <span m="2777310">very</span> <span m="2777510">likely.</span> <span m="2779050">Now,</span> <span m="2779150">you</span> <span m="2779270">want</span> <span m="2779420">to</span> <span m="2779480">recurse</span> <span m="2779900">on</span> <span m="2780030">that?</span> <span m="2781930">I'm</span> <span m="2782080">pretty</span> <span m="2782390">likely</span> <span m="2782710">to</span> <span m="2782770">get</span> <span m="2782900">ahead</span> <span m="2783090">by</span> <span m="2783220">one.</span> </p>
<p><span m="2785490">Well,</span> <span m="2786020">OK,</span> <span m="2786230">get</span> <span m="2786390">ahead</span> <span m="2786560">by</span> <span m="2786660">one.</span> <span m="2786990">I'm</span> <span m="2787140">pretty</span> <span m="2787480">likely</span> <span m="2787810">to</span> <span m="2787890">do</span> <span m="2788070">it</span> <span m="2788120">again.</span> <span m="2789680">And</span> <span m="2789850">I</span> <span m="2789910">did</span> <span m="2790060">it</span> <span m="2790140">again.</span> <span m="2790430">Now I'm</span> <span m="2790570">pretty</span> <span m="2790880">likely</span> <span m="2791200">to</span> <span m="2791290">do</span> <span m="2791460">it</span> <span m="2791520">again.</span> </p>
<p><span m="2792630">And</span> <span m="2792940">there's</span> <span m="2793090">this</span> <span m="2793250">thing</span> <span m="2793410">called</span> <span m="2793680">induction</span> <span m="2793945">that</span> <span m="2794210">we</span> <span m="2794370">worried</span> <span m="2794610">a</span> <span m="2794680">lot</span> <span m="2794900">about.</span> <span m="2795270">So</span> <span m="2795520">by</span> <span m="2795740">induction,</span> <span m="2796200">are we</span> <span m="2796300">likely</span> <span m="2796590">to</span> <span m="2796640">go</span> <span m="2796760">home</span> <span m="2796950">happy</span> <span m="2797230">with</span> <span m="2797360">10?</span> <span m="2798130">No,</span> <span m="2799620">because</span> <span m="2799960">every</span> <span m="2800160">time</span> <span m="2800390">you</span> <span m="2800490">don't</span> <span m="2800870">get</span> <span m="2801100">there,</span> <span m="2801680">you're</span> <span m="2801820">dead.</span> <span m="2803460">You had</span> <span m="2803670">a</span> <span m="2803720">little</span> <span m="2803880">chance</span> <span m="2804170">of</span> <span m="2804310">dying</span> <span m="2804770">and</span> <span m="2804850">not</span> <span m="2805030">reaching</span> <span m="2805380">one,</span> <span m="2805720">and</span> <span m="2805840">a</span> <span m="2805890">little</span> <span m="2806070">chance</span> <span m="2806290">of</span> <span m="2806440">dying</span> <span m="2806830">and</span> <span m="2806900">not</span> <span m="2807080">going</span> <span m="2807250">from</span> <span m="2807340">one</span> <span m="2807540">to</span> <span m="2807620">two.</span> <span m="2808380">And</span> <span m="2808480">you</span> <span m="2808540">add</span> <span m="2808740">up</span> <span m="2808800">all</span> <span m="2808910">those</span> <span m="2809070">chances</span> <span m="2809390">of</span> <span m="2809500">dying,</span> <span m="2810000">and</span> <span m="2810190">you're</span> <span m="2810330">toast,</span> <span m="2811390">because</span> <span m="2811560">that'll</span> <span m="2811810">be</span> <span m="2812130">adding</span> <span m="2812370">up to</span> <span m="2812510">everything,</span> <span m="2813540">pretty</span> <span m="2813740">much.</span> </p>
<p><span m="2816330">So</span> <span m="2816440">that's</span> <span m="2816590">a</span> <span m="2816650">good</span> <span m="2816890">question.</span> <span m="2817400">If</span> <span m="2817500">you're</span> <span m="2817610">likely</span> <span m="2818020">to</span> <span m="2818740">get</span> <span m="2818860">up</span> <span m="2818970">by</span> <span m="2819100">one,</span> <span m="2819410">why</span> <span m="2819620">aren't</span> <span m="2819750">you</span> <span m="2819870">likely</span> <span m="2820150">to</span> <span m="2820200">get</span> <span m="2820340">up</span> <span m="2820440">by</span> <span m="2820560">10?</span> <span m="2822160">It doesn't</span> <span m="2822340">work</span> <span m="2822610">that way.</span> <span m="2822880">That's</span> <span m="2823040">a</span> <span m="2823090">great</span> <span m="2823280">question.</span> </p>
<p><span m="2825300">Let</span> <span m="2825400">me</span> <span m="2825420">show</span> <span m="2825580">you</span> <span m="2825700">the</span> <span m="2825820">phenomenon</span> <span m="2826400">that's</span> <span m="2826590">going</span> <span m="2826830">on</span> <span m="2827090">here,</span> <span m="2832010">as</span> <span m="2832130">to</span> <span m="2832240">why</span> <span m="2833460">it</span> <span m="2833570">works</span> <span m="2833850">out</span> <span m="2834040">this</span> <span m="2834220">way.</span> <span m="2835300">We</span> <span m="2835440">had</span> <span m="2835640">the</span> <span m="2835710">math.</span> <span m="2836800">So</span> <span m="2836830">we</span> <span m="2836930">looked</span> <span m="2837140">at</span> <span m="2837260">it</span> <span m="2837340">that</span> <span m="2837590">way.</span> </p>
<p><span m="2837810">We</span> <span m="2837940">notice</span> <span m="2838340">that</span> <span m="2838940">one</span> <span m="2839170">case</span> <span m="2839460">is</span> <span m="2839580">a</span> <span m="2839650">double</span> <span m="2839930">root</span> <span m="2840170">and</span> <span m="2840230">the</span> <span m="2840320">other</span> <span m="2840480">case</span> <span m="2840810">isn't.</span> <span m="2842200">And</span> <span m="2842340">that</span> <span m="2842490">exponential,</span> <span m="2843160">in</span> <span m="2843210">the</span> <span m="2843310">case</span> <span m="2843640">where</span> <span m="2843940">you</span> <span m="2844220">didn't</span> <span m="2844450">have</span> <span m="2844640">that</span> <span m="2844850">second</span> <span m="2845270">root at 1</span> <span m="2845670">makes</span> <span m="2845930">an</span> <span m="2846000">enormous</span> <span m="2846370">difference.</span> <span m="2848590">Qualitatively,</span> <span m="2849610">we</span> <span m="2849700">can</span> <span m="2849820">draw</span> <span m="2850140">the</span> <span m="2850270">two</span> <span m="2850450">cases.</span> <span m="2852800">So</span> <span m="2853100">in</span> <span m="2853170">the</span> <span m="2853260">case</span> <span m="2853570">of</span> <span m="2853690">an</span> <span m="2853870">unbiased</span> <span m="2854980">or</span> <span m="2855100">fair</span> <span m="2855420">game,</span> <span m="2863010">if</span> <span m="2863240">we</span> <span m="2863340">track</span> <span m="2863710">what's</span> <span m="2863890">going</span> <span m="2864140">on</span> <span m="2864400">over</span> <span m="2864600">time,</span> <span m="2867640">and</span> <span m="2867830">we</span> <span m="2867920">start</span> <span m="2868280">with</span> <span m="2869010">n</span> <span m="2869270">dollars,</span> <span m="2872880">sort</span> <span m="2873250">of</span> <span m="2873440">this</span> <span m="2873650">is</span> <span m="2873790">our</span> <span m="2873950">baseline.</span> </p>
<p><span m="2877350">And</span> <span m="2877560">here's</span> <span m="2877840">our</span> <span m="2877950">target--</span> <span m="2878870">T</span> <span m="2879370">is</span> <span m="2879620">n</span> <span m="2879780">plus</span> <span m="2880050">m.</span> <span m="2882190">And so</span> <span m="2882330">we</span> <span m="2882440">quit</span> <span m="2882720">if</span> <span m="2882850">we</span> <span m="2882950">ever</span> <span m="2883110">get</span> <span m="2883260">here.</span> <span m="2884230">And</span> <span m="2884290">we</span> <span m="2884410">quit</span> <span m="2884720">if</span> <span m="2884850">we</span> <span m="2884950">ever</span> <span m="2885100">hit</span> <span m="2885280">the</span> <span m="2885350">bottom.</span> </p>
<p><span m="2886530">And</span> <span m="2886650">we've</span> <span m="2886740">got</span> <span m="2886890">a</span> <span m="2886960">random</span> <span m="2887380">walk.</span> <span m="2888920">It's</span> <span m="2889980">going</span> <span m="2890230">around,</span> <span m="2893620">just</span> <span m="2893840">doing</span> <span m="2894040">this</span> <span m="2894200">kind</span> <span m="2894370">of</span> <span m="2894440">stuff.</span> <span m="2895200">And</span> <span m="2895390">eventually,</span> <span m="2895980">it's</span> <span m="2896100">going</span> <span m="2896200">to</span> <span m="2896240">hit</span> <span m="2896490">one</span> <span m="2896650">of</span> <span m="2896690">these</span> <span m="2896900">boundaries.</span> <span m="2899150">And</span> <span m="2899290">if</span> <span m="2900040">m</span> <span m="2900320">is</span> <span m="2900420">small</span> <span m="2901110">compared</span> <span m="2901610">to</span> <span m="2901740">n,</span> <span m="2903430">we're</span> <span m="2903580">more</span> <span m="2903830">likely</span> <span m="2904110">to</span> <span m="2904160">hit</span> <span m="2904350">this</span> <span m="2904520">boundary.</span> </p>
<p><span m="2906170">And</span> <span m="2906210">in</span> <span m="2906300">fact,</span> <span m="2906810">the</span> <span m="2906910">chance</span> <span m="2907140">we</span> <span m="2907250">hit</span> <span m="2907440">this</span> <span m="2907640">boundary</span> <span m="2908010">first</span> <span m="2909150">is</span> <span m="2909380">the</span> <span m="2910080">ratio</span> <span m="2910430">of</span> <span m="2910500">these</span> <span m="2910610">sizes.</span> <span m="2911120">It's</span> <span m="2911380">n</span> <span m="2912190">over</span> <span m="2912490">the</span> <span m="2912630">total.</span> <span m="2914230">It's</span> <span m="2914300">the</span> <span m="2914380">chance</span> <span m="2914790">that</span> <span m="2914860">we</span> <span m="2915190">hit</span> <span m="2915420">that</span> <span m="2915690">one</span> <span m="2915860">first.</span> <span m="2918300">Now</span> <span m="2918360">in</span> <span m="2918470">the</span> <span m="2918560">biased</span> <span m="2919120">case,</span> <span m="2920980">the</span> <span m="2921380">picture</span> <span m="2921680">looks</span> <span m="2921900">different.</span> </p>
<p></p>
<p><span m="2935060">So</span> <span m="2935120">in</span> <span m="2935170">the</span> <span m="2935250">biased</span> <span m="2935680">case--</span> <span m="2940200">so</span> <span m="2940360">this</span> <span m="2940550">is</span> <span m="2940670">now</span> <span m="2940840">biased.</span> <span m="2942530">And</span> <span m="2942680">we're</span> <span m="2942810">going</span> <span m="2942910">to</span> <span m="2942990">assume</span> <span m="2943140">it's</span> <span m="2943350">downward</span> <span m="2943860">biased.</span> <span m="2944300">You're</span> <span m="2944350">more</span> <span m="2944540">likely</span> <span m="2944830">to</span> <span m="2944910">lose.</span> </p>
<p><span m="2949960">So</span> <span m="2950070">you</span> <span m="2950180">start at</span> <span m="2950610">n,</span> <span m="2951540">you've</span> <span m="2951640">got</span> <span m="2951780">your</span> <span m="2951880">boundary</span> <span m="2952380">up</span> <span m="2952510">here</span> <span m="2953770">at</span> <span m="2953980">T</span> <span m="2955350">equals</span> <span m="2955670">n</span> <span m="2955820">plus</span> <span m="2956100">m.</span> <span m="2957740">Time</span> <span m="2958260">is</span> <span m="2958420">going</span> <span m="2958690">this</span> <span m="2958890">way.</span> <span m="2961190">The</span> <span m="2961290">problem</span> <span m="2961730">is,</span> <span m="2962260">you've</span> <span m="2962400">got</span> <span m="2962580">a</span> <span m="2963980">downward</span> <span m="2964780">sort</span> <span m="2964950">of</span> <span m="2965060">baseline,</span> <span m="2968810">because</span> <span m="2969470">you</span> <span m="2969610">expect</span> <span m="2970170">to</span> <span m="2970280">lose</span> <span m="2970500">a</span> <span m="2970560">little</span> <span m="2971000">bit</span> <span m="2971320">each</span> <span m="2971610">time.</span> </p>
<p><span m="2973080">And</span> <span m="2973300">so</span> <span m="2973400">you're</span> <span m="2973550">taking</span> <span m="2973930">this</span> <span m="2975060">random</span> <span m="2975500">walk.</span> <span m="2978360">And</span> <span m="2978660">you</span> <span m="2979020">collide</span> <span m="2980830">here.</span> <span m="2981970">And</span> <span m="2983860">these</span> <span m="2984400">things</span> <span m="2984690">are</span> <span m="2984780">known</span> <span m="2985060">as</span> <span m="2985260">the</span> <span m="2985760">swings.</span> <span m="2987050">This</span> <span m="2987410">is</span> <span m="2987620">known</span> <span m="2987930">as</span> <span m="2988410">the</span> <span m="2988510">drift.</span> </p>
<p><span m="2992390">And</span> <span m="2992520">the</span> <span m="2992600">drift</span> <span m="2992950">downward</span> <span m="2993580">is</span> <span m="2993740">1</span> <span m="2993990">minus</span> <span m="2994310">2p.</span> <span m="2995930">That's</span> <span m="2996170">what</span> <span m="2996310">you</span> <span m="2996440">expect</span> <span m="2996920">to</span> <span m="2997010">lose</span> <span m="2997540">if you</span> <span m="2997670">get</span> <span m="2997750">the</span> <span m="2997820">expected</span> <span m="2998510">loss</span> <span m="2999020">on</span> <span m="2999150">each</span> <span m="2999340">bet--</span> <span m="3000630">1</span> <span m="3000820">minus</span> <span m="3001120">2p.</span> <span m="3001690">Because</span> <span m="3001850">you're</span> <span m="3001960">going</span> <span m="3002080">to</span> <span m="3002990">not</span> <span m="3003340">be</span> <span m="3003540">a</span> <span m="3003630">fair</span> <span m="3003930">game.</span> </p>
<p><span m="3004800">This</span> <span m="3005000">one</span> <span m="3005150">has</span> <span m="3005890">zero</span> <span m="3006200">drift</span> <span m="3006500">up</span> <span m="3006620">there.</span> <span m="3006810">It</span> <span m="3006910">stays</span> <span m="3007250">steady.</span> <span m="3008590">And</span> <span m="3008950">in</span> <span m="3009050">random</span> <span m="3009430">walks,</span> <span m="3010930">drift</span> <span m="3012790">outweighs</span> <span m="3016800">the</span> <span m="3016920">swings.</span> </p>
<p><span m="3019900">These</span> <span m="3020110">are</span> <span m="3020150">the</span> <span m="3020280">swings</span> <span m="3020750">here.</span> <span m="3021420">And</span> <span m="3021540">they're</span> <span m="3021710">random.</span> <span m="3023310">The</span> <span m="3023420">drift</span> <span m="3024040">is</span> <span m="3024260">deterministic.</span> <span m="3024980">It's</span> <span m="3025140">steadily</span> <span m="3025650">going</span> <span m="3025930">down.</span> </p>
<p><span m="3026950">And so</span> <span m="3027120">almost</span> <span m="3027590">always</span> <span m="3027940">in</span> <span m="3028030">a</span> <span m="3028080">random</span> <span m="3028410">walk,</span> <span m="3028890">the</span> <span m="3028990">drift</span> <span m="3030250">totally</span> <span m="3030820">takes</span> <span m="3031120">over</span> <span m="3031930">the</span> <span m="3032030">swings.</span> <span m="3032520">The</span> <span m="3032610">swings</span> <span m="3032990">are</span> <span m="3033090">small</span> <span m="3033690">compared</span> <span m="3034070">to</span> <span m="3034140">what</span> <span m="3034280">you're</span> <span m="3034400">losing</span> <span m="3034860">on a</span> <span m="3034930">steady</span> <span m="3035410">basis.</span> <span m="3037730">And</span> <span m="3037830">that's</span> <span m="3037940">why</span> <span m="3038510">you're</span> <span m="3038720">so</span> <span m="3039040">much</span> <span m="3039260">more</span> <span m="3039420">likely</span> <span m="3039710">to</span> <span m="3039790">lose</span> <span m="3041110">when</span> <span m="3041260">you</span> <span m="3041350">have</span> <span m="3041660">the</span> <span m="3043470">drift</span> <span m="3043750">downward.</span> </p>
<p><span m="3046670">Just</span> <span m="3046860">as</span> <span m="3046980">an</span> <span m="3047070">example,</span> <span m="3048110">maybe</span> <span m="3048390">putting</span> <span m="3048680">some</span> <span m="3049310">numbers</span> <span m="3049740">around</span> <span m="3049810">that.</span> <span m="3053680">The</span> <span m="3053810">swings</span> <span m="3054130">are</span> <span m="3054240">the</span> <span m="3054350">same</span> <span m="3054690">in</span> <span m="3054780">both</span> <span m="3055000">cases,</span> <span m="3055700">So</span> <span m="3055990">that</span> <span m="3056420">gives</span> <span m="3056590">you</span> <span m="3056690">some</span> <span m="3057750">qualification</span> <span m="3058620">for</span> <span m="3058770">how</span> <span m="3058900">big</span> <span m="3059180">the</span> <span m="3059260">swings</span> <span m="3059620">tend</span> <span m="3059830">to be.</span> <span m="3063660">We can</span> <span m="3063800">sort of</span> <span m="3063910">do</span> <span m="3064020">that</span> <span m="3064210">with</span> <span m="3064300">standard</span> <span m="3064670">deviation</span> <span m="3065890">notation.</span> </p>
<p><span m="3069380">After</span> <span m="3069890">X</span> <span m="3070335">bets</span> <span m="3071040">or</span> <span m="3071300">X</span> <span m="3071520">steps,</span> <span m="3075520">the</span> <span m="3075620">amount</span> <span m="3075880">you've</span> <span m="3076070">drifted,</span> <span m="3077510">or</span> <span m="3077610">the</span> <span m="3077730">expected</span> <span m="3078400">losses,</span> <span m="3079980">1</span> <span m="3080200">minus</span> <span m="3080550">2p</span> <span m="3081530">X.</span> <span m="3084540">Maybe</span> <span m="3084640">we</span> <span m="3084750">should just</span> <span m="3085470">understand</span> <span m="3085940">why</span> <span m="3086120">this</span> <span m="3086330">is</span> <span m="3086420">the</span> <span m="3086490">case.</span> <span m="3087300">The</span> <span m="3087390">expected</span> <span m="3087980">return</span> <span m="3089450">on</span> <span m="3089540">a</span> <span m="3089630">bet</span> <span m="3094100">is</span> <span m="3094960">1</span> <span m="3095830">with</span> <span m="3095970">probability</span> <span m="3096430">p,</span> <span m="3097960">and</span> <span m="3098150">minus</span> <span m="3098670">1</span> <span m="3098980">with</span> <span m="3099080">probability</span> <span m="3099490">1</span> <span m="3099710">minus</span> <span m="3100070">p.</span> <span m="3101650">And</span> <span m="3101780">so</span> <span m="3101860">that</span> <span m="3102070">is--</span> <span m="3103820">did</span> <span m="3104090">I</span> <span m="3104150">get</span> <span m="3104280">that</span> <span m="3104440">right?</span> <span m="3106046">I think</span> <span m="3106490">that's</span> <span m="3106760">right.</span> </p>
<p><span m="3109730">Oh,</span> <span m="3110030">expected</span> <span m="3110480">loss--</span> <span m="3111190">[INAUDIBLE]</span> <span m="3111330">drifts</span> <span m="3111990">down.</span> <span m="3112200">Instead</span> <span m="3112510">of</span> <span m="3112620">expected</span> <span m="3113000">return,</span> <span m="3113240">let's</span> <span m="3113410">do</span> <span m="3113530">the</span> <span m="3113560">loss,</span> <span m="3114820">because</span> <span m="3114960">that's</span> <span m="3115170">the</span> <span m="3115260">drift.</span> <span m="3115610">It's</span> <span m="3115710">a</span> <span m="3115790">downward</span> <span m="3116320">thing.</span> </p>
<p><span m="3117270">So</span> <span m="3117400">the</span> <span m="3117480">expected</span> <span m="3117970">loss--</span> <span m="3119410">now</span> <span m="3119650">you</span> <span m="3119850">lose</span> <span m="3120360">$1</span> <span m="3120830">with</span> <span m="3120970">1</span> <span m="3121140">minus</span> <span m="3121490">p.</span> <span m="3124610">And</span> <span m="3124780">you</span> <span m="3124880">gain</span> <span m="3125310">$1,</span> <span m="3125700">which</span> <span m="3125880">is</span> <span m="3125990">negative</span> <span m="3126410">loss,</span> <span m="3127670">with</span> <span m="3127790">probability</span> <span m="3128360">p.</span> <span m="3130050">And</span> <span m="3130200">so</span> <span m="3130280">you</span> <span m="3130380">get</span> <span m="3130550">1</span> <span m="3130780">minus</span> <span m="3131130">p</span> <span m="3131380">minus</span> <span m="3131740">p</span> <span m="3132235">is 1</span> <span m="3132730">minus</span> <span m="3133050">2p.</span> </p>
<p><span m="3134570">So</span> <span m="3134690">that's</span> <span m="3134920">your</span> <span m="3135040">expected</span> <span m="3135600">loss.</span> <span m="3136370">Your</span> <span m="3136520">expected</span> <span m="3136960">winnings</span> <span m="3137430">are</span> <span m="3137470">the</span> <span m="3138100">negative</span> <span m="3138600">of</span> <span m="3138720">that.</span> <span m="3141300">So</span> <span m="3141480">after</span> <span m="3141910">x</span> <span m="3142260">steps,</span> <span m="3142830">you</span> <span m="3142980">expect</span> <span m="3143540">to</span> <span m="3143620">lose--</span> <span m="3145550">well,</span> <span m="3145700">I</span> <span m="3145770">just</span> <span m="3145960">add</span> <span m="3146170">up</span> <span m="3146290">the</span> <span m="3146570">linearity</span> <span m="3146940">of</span> <span m="3147000">expectation.</span> </p>
<p><span m="3147700">You</span> <span m="3147800">expect</span> <span m="3148200">to</span> <span m="3148270">lose</span> <span m="3148610">this</span> <span m="3148840">much</span> <span m="3149190">x</span> <span m="3149440">times.</span> <span m="3150950">So</span> <span m="3151090">that's</span> <span m="3151300">your</span> <span m="3151420">expected</span> <span m="3152130">drift.</span> <span m="3152480">You're</span> <span m="3152600">expected</span> <span m="3153120">to</span> <span m="3153200">lose</span> <span m="3153490">that</span> <span m="3153690">much.</span> </p>
<p><span m="3156440">Now,</span> <span m="3156610">the</span> <span m="3156710">swing--</span> <span m="3157660">and</span> <span m="3157760">we</span> <span m="3157840">won't</span> <span m="3158580">prove</span> <span m="3158950">this--</span> <span m="3160520">the</span> <span m="3160640">swing</span> <span m="3161350">is</span> <span m="3161850">expected</span> <span m="3162420">to</span> <span m="3162490">be</span> <span m="3163230">square</span> <span m="3163670">root</span> <span m="3163840">of</span> <span m="3163970">x</span> <span m="3164670">times</span> <span m="3164950">a</span> <span m="3165020">constant.</span> <span m="3165480">So I've</span> <span m="3165630">used</span> <span m="3165870">the</span> <span m="3165980">theta</span> <span m="3166370">notation</span> <span m="3166940">here.</span> <span m="3167670">And</span> <span m="3168150">the</span> <span m="3168290">constant</span> <span m="3168760">is</span> <span m="3168850">small.</span> </p>
<p><span m="3170230">If</span> <span m="3170370">I</span> <span m="3170460">take</span> <span m="3170850">x</span> <span m="3171290">consecutive</span> <span m="3171760">bets</span> <span m="3172050">for</span> <span m="3172160">$1,</span> <span m="3174090">I'm</span> <span m="3174270">very</span> <span m="3174590">likely</span> <span m="3175390">to</span> <span m="3175530">be</span> <span m="3175900">about</span> <span m="3176510">square</span> <span m="3176840">root</span> <span m="3176950">of</span> <span m="3177050">x</span> <span m="3177610">off</span> <span m="3177980">of</span> <span m="3178620">the</span> <span m="3178730">expected</span> <span m="3179240">drift.</span> <span m="3181510">And</span> <span m="3181670">you</span> <span m="3181780">can</span> <span m="3181910">see</span> <span m="3182230">that</span> <span m="3182560">this</span> <span m="3183000">is</span> <span m="3183130">square</span> <span m="3183560">root.</span> <span m="3185040">That</span> <span m="3185420">is</span> <span m="3185570">linear.</span> <span m="3186550">So</span> <span m="3186690">this</span> <span m="3187210">totally</span> <span m="3187860">dominates</span> <span m="3188760">that.</span> </p>
<p><span m="3190250">So</span> <span m="3190390">the</span> <span m="3190520">swings</span> <span m="3191080">are</span> <span m="3191190">generally</span> <span m="3191690">not</span> <span m="3192010">enough</span> <span m="3192260">to</span> <span m="3192370">save</span> <span m="3192730">you.</span> <span m="3194110">And</span> <span m="3194290">so</span> <span m="3194380">you're</span> <span m="3194500">just</span> <span m="3194740">going</span> <span m="3194850">to</span> <span m="3194910">cruise</span> <span m="3195450">downward</span> <span m="3195890">and</span> <span m="3195990">crash,</span> <span m="3197300">almost</span> <span m="3197680">surely.</span> <span m="3201330">OK,</span> <span m="3202050">any</span> <span m="3202220">questions</span> <span m="3204660">about</span> <span m="3204930">that?</span> </p>
<p><span m="3211360">All</span> <span m="3211460">right,</span> <span m="3211540">so</span> <span m="3211620">we</span> <span m="3211710">figured</span> <span m="3212040">out</span> <span m="3212920">the</span> <span m="3213030">probability</span> <span m="3213510">of</span> <span m="3213580">winning</span> <span m="3214160">m</span> <span m="3214430">dollars</span> <span m="3214960">before</span> <span m="3215300">going</span> <span m="3215550">broke.</span> <span m="3216420">That's</span> <span m="3217080">done</span> <span m="3217290">with.</span> <span m="3220230">Now,</span> <span m="3221170">this</span> <span m="3221400">means</span> <span m="3222000">it's</span> <span m="3222190">logical</span> <span m="3222480">to</span> <span m="3222630">conclude</span> <span m="3223060">you're</span> <span m="3223160">likely</span> <span m="3223640">go</span> <span m="3223780">home</span> <span m="3223980">broke in an</span> <span m="3224470">unfair</span> <span m="3224840">game.</span> </p>
<p><span m="3227310">Actually,</span> <span m="3227710">before</span> <span m="3228030">we</span> <span m="3228170">do</span> <span m="3228360">that,</span> <span m="3230030">there's</span> <span m="3230110">one</span> <span m="3230330">other</span> <span m="3230480">case</span> <span m="3230820">we've</span> <span m="3230920">got</span> <span m="3231050">to</span> <span m="3231110">rule</span> <span m="3231400">out.</span> <span m="3232760">We've</span> <span m="3232920">proved</span> <span m="3233500">you're</span> <span m="3233650">likely</span> <span m="3234100">not</span> <span m="3234520">to</span> <span m="3234610">go</span> <span m="3234780">home</span> <span m="3234990">a</span> <span m="3235040">winner.</span> <span m="3236520">Does</span> <span m="3236660">that</span> <span m="3236840">necessarily</span> <span m="3237400">mean</span> <span m="3237630">you're</span> <span m="3237750">likely</span> <span m="3238190">to</span> <span m="3238250">go</span> <span m="3238400">broke?</span> <span m="3238720">I've</span> <span m="3238790">been</span> <span m="3238910">saying</span> <span m="3239290">that,</span> <span m="3239450">but</span> <span m="3239560">there's</span> <span m="3239670">some</span> <span m="3239890">other</span> <span m="3240200">thing</span> <span m="3240380">we</span> <span m="3240470">should</span> <span m="3240650">check.</span> <span m="3243240">What's</span> <span m="3243460">one</span> <span m="3243790">way</span> <span m="3243950">you</span> <span m="3244090">might</span> <span m="3244460">not</span> <span m="3244700">go</span> <span m="3244920">home</span> <span m="3245200">broke?</span> </p>
<p><span m="3247502">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="3248400">PROFESSOR: What</span> <span m="3248560">is</span> <span m="3248670">it?</span> </p>
<p><span m="3249385">AUDIENCE: You don't go home.</span> </p>
<p><span m="3249780">PROFESSOR: You</span> <span m="3249910">don't</span> <span m="3250070">go</span> <span m="3250240">home.</span> <span m="3251220">And</span> <span m="3251370">why</span> <span m="3251530">would</span> <span m="3251660">you</span> <span m="3251740">not</span> <span m="3251960">go</span> <span m="3252100">home?</span> <span m="3253710">Yeah?</span> </p>
<p><span m="3254020">AUDIENCE: You're</span> <span m="3254200">playing forever.</span> </p>
<p><span m="3254680">PROFESSOR: You're</span> <span m="3254860">playing</span> <span m="3255300">forever--</span> <span m="3256810">we</span> <span m="3256990">didn't</span> <span m="3257320">rule</span> <span m="3257650">out</span> <span m="3257810">that</span> <span m="3258010">case--</span> <span m="3258320">you're</span> <span m="3258500">playing</span> <span m="3259030">forever.</span> <span m="3259840">But</span> <span m="3260030">it</span> <span m="3260130">turns</span> <span m="3260480">out,</span> <span m="3261320">if</span> <span m="3261450">you</span> <span m="3261540">did</span> <span m="3261690">the</span> <span m="3261770">same</span> <span m="3262140">analysis,</span> <span m="3262790">you</span> <span m="3262890">can</span> <span m="3263020">analyze</span> <span m="3263360">the</span> <span m="3263440">probability</span> <span m="3263890">of</span> <span m="3263950">going</span> <span m="3264160">home</span> <span m="3264360">broke.</span> <span m="3265200">And</span> <span m="3265310">when</span> <span m="3265400">you</span> <span m="3265530">add</span> <span m="3265920">it to</span> <span m="3266000">the</span> <span m="3266090">probability</span> <span m="3266550">of</span> <span m="3266610">going</span> <span m="3266890">home</span> <span m="3267070">a</span> <span m="3267150">winner,</span> <span m="3267970">it</span> <span m="3268080">adds</span> <span m="3268310">to</span> <span m="3268430">1,</span> <span m="3269220">which</span> <span m="3269500">means</span> <span m="3269740">the</span> <span m="3269830">probability</span> <span m="3270320">playing</span> <span m="3270650">forever</span> <span m="3271360">is</span> <span m="3271630">0.</span> </p>
<p><span m="3273480">Now,</span> <span m="3273600">there</span> <span m="3273780">are</span> <span m="3273970">sample</span> <span m="3274380">points</span> <span m="3274680">where you</span> <span m="3274850">play</span> <span m="3275100">forever.</span> <span m="3276220">But</span> <span m="3276370">when</span> <span m="3276470">you</span> <span m="3276560">add</span> <span m="3276750">up</span> <span m="3276850">all</span> <span m="3277010">those</span> <span m="3277180">sample</span> <span m="3277380">points,</span> <span m="3277590">if</span> <span m="3277680">their</span> <span m="3277800">probability</span> <span m="3278310">is</span> <span m="3278460">0,</span> <span m="3279490">we</span> <span m="3279670">ignore</span> <span m="3280000">them.</span> <span m="3281210">And</span> <span m="3281310">we</span> <span m="3281410">say</span> <span m="3281610">it</span> <span m="3281710">can't</span> <span m="3281960">happen.</span> </p>
<p><span m="3282950">Now,</span> <span m="3283450">we're</span> <span m="3283650">bordering</span> <span m="3284130">on</span> <span m="3284300">philosophy</span> <span m="3284930">here,</span> <span m="3286740">because</span> <span m="3286940">there</span> <span m="3287120">is</span> <span m="3287270">a</span> <span m="3287340">sample</span> <span m="3287690">point here.</span> <span m="3287920">You</span> <span m="3288000">could</span> <span m="3288150">win,</span> <span m="3288330">lose,</span> <span m="3288600">win,</span> <span m="3288770">lose,</span> <span m="3289040">win,</span> <span m="3289160">lose</span> <span m="3289430">forever.</span> <span m="3290640">But</span> <span m="3291300">because</span> <span m="3291740">you</span> <span m="3291830">add</span> <span m="3291970">them</span> <span m="3292090">all</span> <span m="3292240">up</span> <span m="3292400">at</span> <span m="3292520">0,</span> <span m="3293680">measure</span> <span m="3294070">theory</span> <span m="3294470">and some</span> <span m="3294710">math</span> <span m="3294900">we're</span> <span m="3294980">not</span> <span m="3295100">going</span> <span m="3295200">to</span> <span m="3295240">get</span> <span m="3295380">into</span> <span m="3295720">tells</span> <span m="3296040">you</span> <span m="3296430">it</span> <span m="3296510">doesn't</span> <span m="3296770">happen.</span> <span m="3298010">It's</span> <span m="3298400">probability</span> <span m="3298840">1</span> <span m="3299220">you're</span> <span m="3299400">a</span> <span m="3299460">winner</span> <span m="3299790">or a</span> <span m="3299950">loser.</span> <span m="3307430">All</span> <span m="3307580">right,</span> <span m="3307990">so</span> <span m="3308270">I'm</span> <span m="3308400">not</span> <span m="3309230">going</span> <span m="3309320">to</span> <span m="3309380">prove</span> <span m="3309760">that</span> <span m="3310420">the</span> <span m="3310510">probability</span> <span m="3311120">you</span> <span m="3312140">play</span> <span m="3312420">forever</span> <span m="3312850">is</span> <span m="3312980">0.</span> </p>
<p></p>
<p><span m="3324250">But</span> <span m="3324370">let's</span> <span m="3324620">look</span> <span m="3324840">at</span> <span m="3325320">how</span> <span m="3325580">long</span> <span m="3325940">you</span> <span m="3326070">play.</span> <span m="3327660">How</span> <span m="3327910">long</span> <span m="3328550">does</span> <span m="3328690">it</span> <span m="3328820">take</span> <span m="3329060">you</span> <span m="3329170">to</span> <span m="3329260">go</span> <span m="3329510">home</span> <span m="3331040">one</span> <span m="3331230">way or</span> <span m="3331440">another--</span> <span m="3332100">go</span> <span m="3332230">broke?</span> <span m="3333750">And</span> <span m="3334590">to</span> <span m="3334700">do</span> <span m="3334880">this,</span> <span m="3335260">we're</span> <span m="3335360">going</span> <span m="3335370">to</span> <span m="3335430">set</span> <span m="3335660">up</span> <span m="3335800">another</span> <span m="3336150">recurrence.</span> <span m="3339440">So</span> <span m="3339620">we</span> <span m="3339730">know</span> <span m="3339910">eventually</span> <span m="3340540">we</span> <span m="3340670">hit</span> <span m="3340800">a</span> <span m="3340860">boundary.</span> </p>
<p><span m="3342160">I</span> <span m="3342280">want</span> <span m="3342500">to</span> <span m="3342540">know</span> <span m="3343060">how</span> <span m="3343280">many</span> <span m="3343520">bets</span> <span m="3343890">does</span> <span m="3344040">it</span> <span m="3344160">take</span> <span m="3345070">to</span> <span m="3345200">hit</span> <span m="3345430">the</span> <span m="3345520">boundary?</span> <span m="3348100">How</span> <span m="3348290">long</span> <span m="3348700">do</span> <span m="3348810">we</span> <span m="3348910">get</span> <span m="3349070">to</span> <span m="3349150">play</span> <span m="3349420">before</span> <span m="3349700">we</span> <span m="3349820">go</span> <span m="3350000">home</span> <span m="3350210">unhappy?</span> <span m="3353270">So</span> <span m="3353780">S</span> <span m="3354020">will</span> <span m="3354090">be</span> <span m="3354200">the</span> <span m="3354290">number</span> <span m="3354610">of</span> <span m="3354710">steps</span> <span m="3356610">until</span> <span m="3358420">we</span> <span m="3358580">hit</span> <span m="3358800">a</span> <span m="3358850">boundary.</span> </p>
<p><span m="3364320">And</span> <span m="3364480">I</span> <span m="3364570">want</span> <span m="3364770">to</span> <span m="3364830">know</span> <span m="3365130">the</span> <span m="3365310">expected</span> <span m="3366020">number--</span> <span m="3366340">I'll</span> <span m="3366420">call</span> <span m="3366670">it</span> <span m="3366820">E</span> <span m="3366910">sub</span> <span m="3367130">n</span> <span m="3367300">here--</span> <span m="3367570">is</span> <span m="3367660">the</span> <span m="3367780">expected</span> <span m="3368450">value</span> <span m="3368880">of</span> <span m="3368980">S</span> <span m="3370300">given</span> <span m="3370590">that</span> <span m="3370710">I</span> <span m="3370790">start</span> <span m="3371100">with</span> <span m="3371230">n</span> <span m="3371370">dollars.</span> <span m="3375070">I</span> <span m="3375140">mean,</span> <span m="3375550">the</span> <span m="3375650">reason</span> <span m="3375910">you</span> <span m="3376000">could</span> <span m="3376120">think</span> <span m="3376300">about</span> <span m="3376530">this</span> <span m="3376760">is,</span> <span m="3377420">we</span> <span m="3377770">know</span> <span m="3377940">we're</span> <span m="3378040">going</span> <span m="3378140">to</span> <span m="3378190">go</span> <span m="3378360">home</span> <span m="3378550">broke--</span> <span m="3378960">pretty</span> <span m="3379530">likely.</span> <span m="3380410">Do</span> <span m="3380500">we</span> <span m="3380620">at</span> <span m="3380690">least</span> <span m="3380920">have</span> <span m="3381100">some</span> <span m="3381280">fun</span> <span m="3381670">in</span> <span m="3381760">the</span> <span m="3381820">meantime?</span> <span m="3382810">Do we</span> <span m="3382970">get</span> <span m="3383080">a</span> <span m="3383140">lot</span> <span m="3383490">gambling</span> <span m="3383950">in</span> <span m="3384265">and</span> <span m="3384580">free</span> <span m="3384810">drinks,</span> <span m="3385200">or</span> <span m="3385280">whatever,</span> <span m="3385640">before</span> <span m="3386840">we're</span> <span m="3388100">killed</span> <span m="3388390">here?</span> </p>
<p><span m="3391590">Now,</span> <span m="3391890">this</span> <span m="3392120">also</span> <span m="3392450">has</span> <span m="3392660">a</span> <span m="3392720">recurrence.</span> <span m="3393210">And</span> <span m="3393300">I'm</span> <span m="3393380">going</span> <span m="3393490">to</span> <span m="3394120">show</span> <span m="3394340">you</span> <span m="3394470">what</span> <span m="3394580">it</span> <span m="3394680">is,</span> <span m="3394890">then</span> <span m="3395030">prove</span> <span m="3395350">that</span> <span m="3395500">that's</span> <span m="3396490">correct.</span> <span m="3397940">So</span> <span m="3398130">I</span> <span m="3398220">claim</span> <span m="3398690">that</span> <span m="3399800">the</span> <span m="3401870">expected</span> <span m="3402360">number</span> <span m="3402610">of</span> <span m="3402680">steps</span> <span m="3403120">given</span> <span m="3403320">we</span> <span m="3403420">start</span> <span m="3403670">with</span> <span m="3403790">n</span> <span m="3403930">dollars</span> <span m="3404600">is</span> <span m="3405930">0</span> <span m="3406490">if</span> <span m="3406630">we</span> <span m="3406720">start</span> <span m="3407080">with</span> <span m="3407200">no</span> <span m="3407360">money,</span> <span m="3409220">because</span> <span m="3409520">we</span> <span m="3409650">are</span> <span m="3409750">already</span> <span m="3410020">broke.</span> </p>
<p><span m="3411500">It's</span> <span m="3411750">0</span> <span m="3413000">if</span> <span m="3413200">we</span> <span m="3413320">start</span> <span m="3414960">with</span> <span m="3415430">T</span> <span m="3415760">dollars,</span> <span m="3417370">because</span> <span m="3417530">then</span> <span m="3417680">we</span> <span m="3417760">just</span> <span m="3417930">go</span> <span m="3418070">home</span> <span m="3418770">happy.</span> <span m="3419110">There's</span> <span m="3419300">no</span> <span m="3419460">bets,</span> <span m="3419750">because</span> <span m="3419890">we've</span> <span m="3420070">already</span> <span m="3420350">hit</span> <span m="3420600">the</span> <span m="3420730">upper</span> <span m="3420980">boundary.</span> <span m="3423290">And</span> <span m="3423450">the</span> <span m="3423570">interesting</span> <span m="3423920">case</span> <span m="3424320">will</span> <span m="3424430">be</span> <span m="3424730">it's</span> <span m="3424900">1</span> <span m="3425240">plus</span> <span m="3425750">p</span> <span m="3426140">times</span> <span m="3427265">E</span> <span m="3427750">n</span> <span m="3428240">minus</span> <span m="3428590">1</span> <span m="3429490">plus</span> <span m="3429870">1</span> <span m="3430110">minus</span> <span m="3430470">p--</span> <span m="3431980">oops,</span> <span m="3433435">n</span> <span m="3433920">plus</span> <span m="3434220">1--</span> <span m="3435300">1</span> <span m="3435450">minus</span> <span m="3435730">p</span> <span m="3436105">E n</span> <span m="3436480">minus</span> <span m="3436860">1,</span> <span m="3437980">if</span> <span m="3438200">we</span> <span m="3438290">start</span> <span m="3438850">with</span> <span m="3439020">between</span> <span m="3439390">0</span> <span m="3440570">and</span> <span m="3440780">T</span> <span m="3440960">dollars.</span> </p>
<p><span m="3446980">OK,</span> <span m="3448920">so</span> <span m="3449090">let's</span> <span m="3449860">prove</span> <span m="3450260">that.</span> <span m="3453400">Actually,</span> <span m="3454050">the</span> <span m="3454150">proof</span> <span m="3456430">is</span> <span m="3456980">exactly</span> <span m="3457500">the</span> <span m="3457620">same</span> <span m="3457990">as</span> <span m="3458080">the</span> <span m="3458140">last</span> <span m="3458510">one.</span> <span m="3458660">So</span> <span m="3458790">I</span> <span m="3458850">don't</span> <span m="3459010">think</span> <span m="3459150">I</span> <span m="3459250">need to</span> <span m="3459722">do it.</span> </p>
<p><span m="3464300">The</span> <span m="3464390">proof</span> <span m="3464720">is</span> <span m="3465730">pretty</span> <span m="3465990">simple,</span> <span m="3466840">because</span> <span m="3467280">we</span> <span m="3467420">look</span> <span m="3467620">at</span> <span m="3467710">two</span> <span m="3467850">cases.</span> <span m="3468370">You</span> <span m="3468520">win</span> <span m="3468790">the</span> <span m="3468880">first</span> <span m="3469200">bet--</span> <span m="3470790">happens</span> <span m="3471160">with</span> <span m="3471270">probability</span> <span m="3471790">p.</span> <span m="3472150">And</span> <span m="3472220">then</span> <span m="3472350">you're</span> <span m="3472490">starting</span> <span m="3472890">with</span> <span m="3473030">n</span> <span m="3473140">plus</span> <span m="3473370">$1</span> <span m="3473940">over</span> <span m="3474190">again.</span> <span m="3475250">Or</span> <span m="3476310">you</span> <span m="3476480">lose</span> <span m="3476790">the</span> <span m="3476890">first</span> <span m="3477200">bet--</span> <span m="3477690">happens</span> <span m="3478050">with</span> <span m="3478130">probability</span> <span m="3478530">1</span> <span m="3478750">minus</span> <span m="3479090">p.</span> <span m="3480360">And</span> <span m="3481900">you're</span> <span m="3482050">starting</span> <span m="3482380">over</span> <span m="3482560">with</span> <span m="3482730">n</span> <span m="3482870">minus</span> <span m="3483270">$1</span> <span m="3483890">now--</span> <span m="3484520">same</span> <span m="3484880">as</span> <span m="3484980">last</span> <span m="3485310">time.</span> </p>
<p><span m="3485920">In</span> <span m="3486020">fact,</span> <span m="3486100">this</span> <span m="3486250">whole</span> <span m="3486480">recurrence</span> <span m="3486960">is</span> <span m="3487080">identical</span> <span m="3487590">to</span> <span m="3487650">last</span> <span m="3488020">time</span> <span m="3489100">except</span> <span m="3489450">for</span> <span m="3489540">one</span> <span m="3489790">thing.</span> <span m="3491690">What's</span> <span m="3491870">the</span> <span m="3491960">one</span> <span m="3492130">thing</span> <span m="3492290">that's</span> <span m="3492430">different</span> <span m="3492720">now?</span> </p>
<p><span m="3493914">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="3494870">PROFESSOR: What</span> <span m="3495020">is</span> <span m="3495130">it?</span> </p>
<p><span m="3495994">AUDIENCE: You have</span> <span m="3496458">[INAUDIBLE].</span> </p>
<p><span m="3498780">PROFESSOR: You</span> <span m="3499020">have--</span> </p>
<p><span m="3500738">AUDIENCE: So it's</span> <span m="3501232">not</span> <span m="3501726">[INAUDIBLE]</span> <span m="3502220">any more.</span> </p>
<p><span m="3503208">PROFESSOR: That's  different.</span> <span m="3504690">There's</span> <span m="3504850">another</span> <span m="3505165">difference.</span> <span m="3506920">That's</span> <span m="3507400">one</span> <span m="3507640">difference</span> <span m="3508740">that's</span> <span m="3508930">going</span> <span m="3509040">to</span> <span m="3509090">make</span> <span m="3509320">it</span> <span m="3509450">inhomogeneous.</span> <span m="3510100">That's</span> <span m="3510550">sort</span> <span m="3510740">of</span> <span m="3510800">a</span> <span m="3510870">pain.</span> </p>
<p><span m="3511770">What's</span> <span m="3511950">the</span> <span m="3512050">other</span> <span m="3512190">difference from</span> <span m="3512560">last</span> <span m="3512870">time?</span> <span m="3513200">This</span> <span m="3513460">part's</span> <span m="3513650">the</span> <span m="3513730">same</span> <span m="3513980">otherwise.</span> </p>
<p><span m="3515146">AUDIENCE: Boundaries.</span> </p>
<p><span m="3515900">PROFESSOR: What</span> <span m="3516050">is</span> <span m="3516070">it?</span> </p>
<p><span m="3516400">AUDIENCE: Boundary</span> <span m="3516810">conditions.</span> </p>
<p><span m="3517220">PROFESSOR: Boundary</span> <span m="3517620">conditions--</span> <span m="3519190">that</span> <span m="3519380">was</span> <span m="3519480">a</span> <span m="3519560">1</span> <span m="3519830">before.</span> <span m="3520120">Now it's a</span> <span m="3520500">0.</span> <span m="3522290">OK,</span> <span m="3523220">so</span> <span m="3523370">a</span> <span m="3523460">little</span> <span m="3523740">change</span> <span m="3524100">here,</span> <span m="3524360">and</span> <span m="3524430">I</span> <span m="3524500">added</span> <span m="3524800">a</span> <span m="3524870">1</span> <span m="3525090">here.</span> <span m="3526420">But</span> <span m="3526730">that's</span> <span m="3526980">going</span> <span m="3527080">to</span> <span m="3527130">make</span> <span m="3527310">it</span> <span m="3528000">a</span> <span m="3528240">pretty</span> <span m="3528440">different</span> <span m="3528750">answer.</span> </p>
<p><span m="3530180">So</span> <span m="3530320">let's</span> <span m="3530540">see</span> <span m="3530670">what</span> <span m="3530820">the</span> <span m="3530900">recurrence</span> <span m="3531390">is.</span> <span m="3531540">I'll</span> <span m="3531690">rearrange</span> <span m="3534540">terms</span> <span m="3534880">here</span> <span m="3535000">to</span> <span m="3535070">put</span> <span m="3535250">it into</span> <span m="3535460">recurrence.</span> <span m="3537310">I</span> <span m="3537400">get</span> <span m="3537730">p</span> <span m="3538530">E</span> <span m="3538790">sub</span> <span m="3539070">n</span> <span m="3539360">plus</span> <span m="3539710">1</span> <span m="3540890">minus</span> <span m="3542030">E n</span> <span m="3543460">plus</span> <span m="3543810">1</span> <span m="3544030">minus</span> <span m="3544390">p</span> <span m="3545610">E</span> <span m="3545960">n</span> <span m="3546100">minus</span> <span m="3546530">1</span> <span m="3547650">equals</span> <span m="3548370">minus</span> <span m="3548800">1,</span> <span m="3549210">not</span> <span m="3549440">0.</span> <span m="3550710">And</span> <span m="3551130">the</span> <span m="3551190">boundary</span> <span m="3551570">conditions</span> <span m="3552490">are</span> <span m="3552830">E 0</span> <span m="3553260">is</span> <span m="3553370">0</span> <span m="3554440">and E T</span> <span m="3554920">is</span> <span m="3555130">0.</span> </p>
<p><span m="3558210">OK,</span> <span m="3560460">what's</span> <span m="3560600">the</span> <span m="3560720">first</span> <span m="3561070">thing</span> <span m="3561210">you</span> <span m="3561290">do</span> <span m="3561550">when</span> <span m="3561670">you</span> <span m="3561760">have</span> <span m="3562000">an</span> <span m="3562150">inhomogeneous</span> <span m="3563400">linear</span> <span m="3563810">recurrence?</span> <span m="3566020">Solve</span> <span m="3566430">the</span> <span m="3566510">homogeneous</span> <span m="3567560">one.</span> <span m="3568920">And</span> <span m="3569460">the</span> <span m="3569620">answer</span> <span m="3569940">there--</span> <span m="3570320">well,</span> <span m="3570500">it's</span> <span m="3570650">the</span> <span m="3570730">same</span> <span m="3571050">as</span> <span m="3571590">before.</span> <span m="3572050">This</span> <span m="3572290">is</span> <span m="3572430">the</span> <span m="3572540">part</span> <span m="3572780">we</span> <span m="3572890">analyzed.</span> </p>
<p><span m="3573990">And</span> <span m="3574210">we'll</span> <span m="3574360">do</span> <span m="3574540">it</span> <span m="3574620">for</span> <span m="3574690">the</span> <span m="3574800">case</span> <span m="3575160">when</span> <span m="3575320">p</span> <span m="3575570">is</span> <span m="3576390">not</span> <span m="3576650">1/2--</span> <span m="3577000">so</span> <span m="3577150">the</span> <span m="3577320">unfair</span> <span m="3577790">game.</span> <span m="3579770">So</span> <span m="3580760">the</span> <span m="3580820">homogeneous</span> <span m="3581500">solution</span> <span m="3586310">is</span> <span m="3586690">E n</span> <span m="3587270">just</span> <span m="3587640">from before--</span> <span m="3588020">same</span> <span m="3588330">thing--</span> <span m="3590130">1</span> <span m="3590280">minus</span> <span m="3590600">p</span> <span m="3590800">over</span> <span m="3590980">p</span> <span m="3591870">to</span> <span m="3591910">the</span> <span m="3592020">n</span> <span m="3592270">plus</span> <span m="3592580">B.</span> <span m="3593880">And</span> <span m="3593990">this</span> <span m="3594150">is</span> <span m="3594260">the</span> <span m="3594370">case</span> <span m="3594780">with</span> <span m="3594930">two</span> <span m="3595200">roots.</span> <span m="3595720">p</span> <span m="3595980">does not</span> <span m="3596370">equal 1/2.</span> </p>
<p><span m="3600620">What's</span> <span m="3600890">the</span> <span m="3600970">next</span> <span m="3601220">thing</span> <span m="3601360">you</span> <span m="3601450">do</span> <span m="3603780">for</span> <span m="3604250">inhomogeneous</span> <span m="3605170">recurrence?</span> <span m="3607634">Are</span> <span m="3608100">we</span> <span m="3608310">plugging in</span> <span m="3608700">boundary</span> <span m="3609040">conditions</span> <span m="3609490">yet?</span> <span m="3611710">No.</span> <span m="3612120">So</span> <span m="3612250">what</span> <span m="3612320">do</span> <span m="3612390">I</span> <span m="3612460">do</span> <span m="3612560">next?</span> <span m="3614840">Particular</span> <span m="3615540">solution.</span> </p>
<p><span m="3621160">And</span> <span m="3622020">what's</span> <span m="3622370">my</span> <span m="3622540">first</span> <span m="3622920">guess?</span> <span m="3625660">We</span> <span m="3625810">have</span> <span m="3625830">the</span> <span m="3626316">recurrence</span> <span m="3630690">like</span> <span m="3630950">this</span> <span m="3631350">here.</span> <span m="3634590">What</span> <span m="3634720">do</span> <span m="3634820">I</span> <span m="3634900">guess</span> <span m="3635270">for</span> <span m="3635460">E</span> <span m="3635930">n?</span> <span m="3637960">I'm</span> <span m="3638240">trying</span> <span m="3638610">to</span> <span m="3638680">guess</span> <span m="3638975">something</span> <span m="3639270">that</span> <span m="3639480">looks</span> <span m="3639700">like</span> <span m="3639930">that.</span> <span m="3642610">So</span> <span m="3642670">what</span> <span m="3642780">do</span> <span m="3642870">I</span> <span m="3642960">guess?</span> </p>
<p><span m="3645380">Constant,</span> <span m="3646320">yeah.</span> <span m="3647740">That's</span> <span m="3648000">a</span> <span m="3648050">scalar.</span> <span m="3648420">I</span> <span m="3648520">just</span> <span m="3648720">guess</span> <span m="3648930">a</span> <span m="3648980">constant.</span> <span m="3651120">And</span> <span m="3651860">if</span> <span m="3652020">I</span> <span m="3652130">plug</span> <span m="3652440">a</span> <span m="3652750">constant</span> <span m="3653035">a</span> <span m="3653320">into</span> <span m="3654040">here,</span> <span m="3655480">it's</span> <span m="3655730">going</span> <span m="3655830">to</span> <span m="3655890">fail.</span> <span m="3657420">Because</span> <span m="3657870">I'll</span> <span m="3658000">just</span> <span m="3659990">pull</span> <span m="3660250">the a</span> <span m="3660430">out.</span> <span m="3661730">I'll</span> <span m="3661880">get</span> <span m="3662230">p</span> <span m="3662950">minus</span> <span m="3663350">1</span> <span m="3663860">plus</span> <span m="3664160">1</span> <span m="3664440">minus</span> <span m="3664820">p</span> <span m="3665100">is</span> <span m="3665280">0,</span> <span m="3665740">and</span> <span m="3665850">0</span> <span m="3666130">doesn't</span> <span m="3666380">equal</span> <span m="3666600">minus</span> <span m="3666930">1.</span> <span m="3667640">So</span> <span m="3667810">it</span> <span m="3667920">fails.</span> </p>
<p><span m="3672240">So</span> <span m="3672460">I</span> <span m="3672540">guess</span> <span m="3672830">again.</span> <span m="3673290">What</span> <span m="3673330">do</span> <span m="3673410">I</span> <span m="3673490">guess</span> <span m="3673790">next</span> <span m="3674060">time?</span> <span m="3676380">a n</span> <span m="3676840">plus</span> <span m="3677160">b.</span> <span m="3679990">All</span> <span m="3680480">right,</span> <span m="3680800">and</span> <span m="3682030">I</span> <span m="3682170">don't</span> <span m="3682330">think</span> <span m="3682470">I'll</span> <span m="3682580">drag</span> <span m="3683050">you</span> <span m="3683170">through</span> <span m="3683390">all</span> <span m="3683700">the</span> <span m="3684600">algebra</span> <span m="3685010">for</span> <span m="3685120">that,</span> <span m="3685530">but</span> <span m="3685640">it</span> <span m="3685740">works.</span> </p>
<p><span m="3688430">And</span> <span m="3689060">when</span> <span m="3689280">you</span> <span m="3689870">do</span> <span m="3690100">it,</span> <span m="3690280">you</span> <span m="3690420">find</span> <span m="3690900">that</span> <span m="3693400">a</span> <span m="3693900">is</span> <span m="3694270">minus</span> <span m="3694820">1</span> <span m="3695310">over</span> <span m="3696410">2p</span> <span m="3697030">minus</span> <span m="3697440">1.</span> <span m="3698100">And</span> <span m="3698280">b</span> <span m="3698510">could</span> <span m="3698630">be</span> <span m="3698780">anything.</span> <span m="3699470">So</span> <span m="3700780">let</span> <span m="3701120">me</span> <span m="3701250">just</span> <span m="3701400">rewrite</span> <span m="3701750">this</span> <span m="3701920">as</span> <span m="3702030">1</span> <span m="3702270">over</span> <span m="3702420">1</span> <span m="3702580">minus</span> <span m="3702860">2p.</span> <span m="3704080">And</span> <span m="3704220">b</span> <span m="3704430">can</span> <span m="3704550">be</span> <span m="3704660">anything,</span> <span m="3705100">so</span> <span m="3705120">we'll</span> <span m="3705210">set</span> <span m="3705480">b</span> <span m="3705650">equal</span> <span m="3705840">to</span> <span m="3705900">0.</span> </p>
<p><span m="3709490">So</span> <span m="3709620">we've</span> <span m="3709720">got</span> <span m="3709900">our</span> <span m="3709990">particular</span> <span m="3710470">solution.</span> <span m="3712910">It's</span> <span m="3713090">not</span> <span m="3713250">hard</span> <span m="3713540">to</span> <span m="3713590">go</span> <span m="3713770">compute</span> <span m="3714120">that.</span> <span m="3714520">You just</span> <span m="3714885">plug</span> <span m="3715250">it</span> <span m="3715330">back</span> <span m="3715580">in</span> <span m="3716918">and solve.</span> <span m="3724670">Now</span> <span m="3724900">we</span> <span m="3725050">add</span> <span m="3725240">them</span> <span m="3725350">together</span> <span m="3725830">to</span> <span m="3725890">get</span> <span m="3726050">the</span> <span m="3726140">general</span> <span m="3726570">solution.</span> </p>
<p></p>
<p><span m="3738830">This</span> <span m="3739050">is</span> <span m="3739210">A n</span> <span m="3739670">plus</span> <span m="3739900">B.</span> <span m="3740180">B</span> <span m="3740460">was</span> <span m="3740640">0,</span> <span m="3741080">and</span> <span m="3741150">here's</span> <span m="3741440">A as</span> <span m="3741800">1</span> <span m="3742020">over</span> <span m="3743070">1</span> <span m="3743250">minus</span> <span m="3743540">2p.</span> <span m="3745440">And</span> <span m="3745620">now</span> <span m="3745900">what</span> <span m="3746020">do</span> <span m="3746070">we</span> <span m="3746200">do</span> <span m="3748880">to</span> <span m="3748970">finish?</span> <span m="3749270">I've</span> <span m="3749370">got</span> <span m="3749510">my</span> <span m="3749620">general</span> <span m="3750000">solution</span> <span m="3750470">here</span> <span m="3753700">by</span> <span m="3753830">adding</span> <span m="3754170">up</span> <span m="3754290">the</span> <span m="3754380">homogeneous</span> <span m="3754720">and</span> <span m="3755060">the</span> <span m="3755170">particular</span> <span m="3755580">solution.</span> <span m="3757810">Plug in</span> <span m="3758040">the</span> <span m="3758110">boundary</span> <span m="3758490">conditions.</span> </p>
<p><span m="3765680">All</span> <span m="3765890">right,</span> <span m="3766500">I'm</span> <span m="3766740">not</span> <span m="3766910">going</span> <span m="3767010">to</span> <span m="3767090">drag</span> <span m="3767590">you</span> <span m="3767690">through</span> <span m="3768130">solving</span> <span m="3768620">this</span> <span m="3769630">case,</span> <span m="3770100">but</span> <span m="3770260">I'm</span> <span m="3770360">going</span> <span m="3770470">to</span> <span m="3770530">show</span> <span m="3770740">you</span> <span m="3770860">the</span> <span m="3771040">answer.</span> <span m="3775430">E n</span> <span m="3775900">equals</span> <span m="3776480">n</span> <span m="3776970">over</span> <span m="3777330">1</span> <span m="3777590">minus</span> <span m="3777950">2p</span> <span m="3780650">minus</span> <span m="3782090">T,</span> <span m="3782400">the</span> <span m="3782530">upper</span> <span m="3782710">boundary,</span> <span m="3783480">over</span> <span m="3783680">1</span> <span m="3783910">minus</span> <span m="3784270">2p</span> <span m="3786090">times</span> <span m="3787090">1</span> <span m="3787310">minus</span> <span m="3787660">p</span> <span m="3788070">over</span> <span m="3789080">p</span> <span m="3789440">to</span> <span m="3789510">the</span> <span m="3789840">n</span> <span m="3790500">minus</span> <span m="3790920">1</span> <span m="3791440">over</span> <span m="3793220">1</span> <span m="3793400">minus</span> <span m="3793810">p</span> <span m="3794260">over</span> <span m="3794560">p</span> <span m="3795670">to</span> <span m="3795790">the</span> <span m="3795960">T</span> <span m="3796260">minus</span> <span m="3796670">1.</span> <span m="3797930">So</span> <span m="3798080">actually,</span> <span m="3798410">this</span> <span m="3798630">looks</span> <span m="3798810">a</span> <span m="3798860">little</span> <span m="3799130">familiar</span> <span m="3799570">from</span> <span m="3799730">the</span> <span m="3799810">last</span> <span m="3800160">time</span> <span m="3800670">when</span> <span m="3800800">we</span> <span m="3800880">did</span> <span m="3801040">this</span> <span m="3801200">recurrence,</span> <span m="3802720">figuring</span> <span m="3802970">out</span> <span m="3803090">the</span> <span m="3803180">probability</span> <span m="3803760">we</span> <span m="3803850">go</span> <span m="3803980">home</span> <span m="3804100">a</span> <span m="3804170">winner.</span> </p>
<p><span m="3804450">Here</span> <span m="3804730">this</span> <span m="3804930">is</span> <span m="3805000">the</span> <span m="3805120">expected</span> <span m="3805760">number</span> <span m="3806000">of</span> <span m="3806080">steps</span> <span m="3807630">to</span> <span m="3807760">hit</span> <span m="3807930">a</span> <span m="3807980">boundary,</span> <span m="3808790">to</span> <span m="3808960">go</span> <span m="3809180">home.</span> <span m="3810900">If</span> <span m="3811010">we</span> <span m="3811170">plug in</span> <span m="3811670">the</span> <span m="3811760">values,</span> <span m="3812480">it's</span> <span m="3812610">a</span> <span m="3812650">little</span> <span m="3812810">hairy,</span> <span m="3813130">but</span> <span m="3813270">you</span> <span m="3813370">can</span> <span m="3813520">compute</span> <span m="3813890">it.</span> <span m="3815850">So</span> <span m="3816160">for</span> <span m="3816300">example,</span> <span m="3817800">if</span> <span m="3818070">m</span> <span m="3818325">is</span> <span m="3818580">100,</span> <span m="3820700">n</span> <span m="3821010">is</span> <span m="3821160">1,000,</span> <span m="3822860">T</span> <span m="3823660">would</span> <span m="3823780">be</span> <span m="3823910">1,100</span> <span m="3824810">in</span> <span m="3824920">that</span> <span m="3825110">case.</span> <span m="3827160">p</span> <span m="3827470">is</span> <span m="3827620">9/19</span> <span m="3829280">playing</span> <span m="3829830">roulette.</span> </p>
<p><span m="3832210">Then</span> <span m="3833250">the</span> <span m="3833360">expected</span> <span m="3833940">number</span> <span m="3834230">of</span> <span m="3834370">bets</span> <span m="3836570">before</span> <span m="3836910">you</span> <span m="3837370">have</span> <span m="3837550">to</span> <span m="3837630">go</span> <span m="3837780">home</span> <span m="3839580">is</span> <span m="3841400">1,900</span> <span m="3842060">from</span> <span m="3842310">this</span> <span m="3842580">part,</span> <span m="3846060">minus</span> <span m="3847450">0.56</span> <span m="3850130">from</span> <span m="3850310">that</span> <span m="3850550">part.</span> <span m="3853480">So</span> <span m="3855100">actually</span> <span m="3855400">19,000,</span> <span m="3856350">sorry.</span> <span m="3857570">So</span> <span m="3857850">it's</span> <span m="3858650">very</span> <span m="3858890">close</span> <span m="3859210">to</span> <span m="3859310">19,000</span> <span m="3862150">bets</span> <span m="3862370">you've</span> <span m="3862500">got</span> <span m="3862650">to</span> <span m="3862700">make.</span> </p>
<p><span m="3864580">So</span> <span m="3864900">it</span> <span m="3864980">takes</span> <span m="3865190">a</span> <span m="3865260">long</span> <span m="3865750">time</span> <span m="3866210">to</span> <span m="3866330">lose</span> <span m="3866600">$1,000.</span> <span m="3868230">And</span> <span m="3869960">it</span> <span m="3870180">sort</span> <span m="3870460">of</span> <span m="3870740">comes</span> <span m="3871010">very</span> <span m="3871470">close</span> <span m="3871840">to</span> <span m="3871940">the</span> <span m="3872100">answer</span> <span m="3872410">you</span> <span m="3872560">would</span> <span m="3872670">have</span> <span m="3872820">guessed</span> <span m="3873350">without</span> <span m="3873990">thinking</span> <span m="3874400">and</span> <span m="3874510">solving</span> <span m="3874860">the</span> <span m="3874940">recurrence.</span> <span m="3876120">If</span> <span m="3876260">you</span> <span m="3876400">expect</span> <span m="3876910">to</span> <span m="3877050">lose</span> <span m="3878130">1</span> <span m="3878510">minus</span> <span m="3878880">2p</span> <span m="3879460">every</span> <span m="3879720">bet,</span> <span m="3881350">and</span> <span m="3881490">you</span> <span m="3881580">want</span> <span m="3881770">to</span> <span m="3881810">know</span> <span m="3882230">how</span> <span m="3882440">long</span> <span m="3882850">the</span> <span m="3882910">expected</span> <span m="3883330">time</span> <span m="3883640">to</span> <span m="3883750">lose</span> <span m="3884160">n</span> <span m="3884410">dollars,</span> <span m="3885860">you</span> <span m="3886010">might</span> <span m="3886240">well</span> <span m="3886500">have</span> <span m="3886570">said,</span> <span m="3886890">I</span> <span m="3886940">think</span> <span m="3887170">it's</span> <span m="3887320">going</span> <span m="3887420">to</span> <span m="3887490">be</span> <span m="3887690">n</span> <span m="3888010">over</span> <span m="3888250">the</span> <span m="3888560">amount</span> <span m="3888850">I</span> <span m="3888910">lose</span> <span m="3889200">every</span> <span m="3889410">time.</span> </p>
<p><span m="3891210">That</span> <span m="3891380">would</span> <span m="3891500">be</span> <span m="3891620">wrong,</span> <span m="3892900">technically,</span> <span m="3894460">because</span> <span m="3894620">you'd</span> <span m="3894730">have</span> <span m="3894910">left</span> <span m="3895230">off</span> <span m="3895450">this</span> <span m="3895760">nasty</span> <span m="3896170">thing.</span> <span m="3897400">But</span> <span m="3897520">this</span> <span m="3897720">nasty</span> <span m="3898050">thing</span> <span m="3898480">doesn't</span> <span m="3899080">make</span> <span m="3899690">much</span> <span m="3899930">of</span> <span m="3899990">a</span> <span m="3900120">real</span> <span m="3900410">difference,</span> <span m="3900780">because</span> <span m="3900940">it</span> <span m="3901080">goes</span> <span m="3901370">to</span> <span m="3901460">0</span> <span m="3901970">really</span> <span m="3902240">fast</span> <span m="3902810">for</span> <span m="3902920">any</span> <span m="3903050">numbers</span> <span m="3903400">like</span> <span m="3903790">100</span> <span m="3904245">and</span> <span m="3904700">1,000--</span> <span m="3905430">makes</span> <span m="3905670">no</span> <span m="3905800">difference</span> <span m="3906200">at</span> <span m="3906260">all.</span> <span m="3906570">So</span> <span m="3906620">the</span> <span m="3906750">intuition</span> <span m="3907060">in</span> <span m="3907370">that</span> <span m="3907590">case</span> <span m="3907860">comes</span> <span m="3908130">out</span> <span m="3908380">to</span> <span m="3908430">be</span> <span m="3908560">pretty</span> <span m="3908850">close,</span> <span m="3910440">even</span> <span m="3910650">though</span> <span m="3910770">technically,</span> <span m="3911290">it's</span> <span m="3911430">not</span> <span m="3911590">exactly</span> <span m="3912950">right.</span> </p>
<p><span m="3915540">Now,</span> <span m="3915680">to</span> <span m="3915770">see</span> <span m="3915980">why</span> <span m="3916120">this</span> <span m="3916370">goes</span> <span m="3916620">to</span> <span m="3916720">0,</span> <span m="3919070">if</span> <span m="3919260">T</span> <span m="3919870">equals</span> <span m="3920080">n</span> <span m="3920250">plus</span> <span m="3920540">m</span> <span m="3920780">here--</span> <span m="3921460">this</span> <span m="3921660">is</span> <span m="3921780">n</span> <span m="3921920">plus</span> <span m="3922210">m--</span> <span m="3924170">and</span> <span m="3924500">your</span> <span m="3924650">upper</span> <span m="3924900">limits,</span> <span m="3925350">say</span> <span m="3925610">m</span> <span m="3925870">goes</span> <span m="3926820">to</span> <span m="3926950">infinity--</span> <span m="3928010">it's</span> <span m="3928160">100</span> <span m="3928600">in</span> <span m="3928670">this</span> <span m="3928850">case--</span> <span m="3930110">then</span> <span m="3931040">that</span> <span m="3931280">just</span> <span m="3931700">zooms</span> <span m="3932110">to</span> <span m="3932200">0,</span> <span m="3933400">and</span> <span m="3933530">you're</span> <span m="3933690">only</span> <span m="3933990">left</span> <span m="3934380">with</span> <span m="3934480">that.</span> <span m="3934770">Which</span> <span m="3934970">means</span> <span m="3935340">that</span> <span m="3935570">we</span> <span m="3935680">can</span> <span m="3935770">use</span> <span m="3935960">asymptotic</span> <span m="3936510">notation</span> <span m="3937020">here</span> <span m="3939510">to</span> <span m="3939660">sort of</span> <span m="3939760">characterize</span> <span m="3941035">the</span> <span m="3941320">expected</span> <span m="3941575">number</span> <span m="3941830">of</span> <span m="3942306">bets.</span> <span m="3948020">And</span> <span m="3948140">it's</span> <span m="3948320">totally</span> <span m="3948780">dominated</span> <span m="3949300">by</span> <span m="3949420">the</span> <span m="3949530">drift.</span> <span m="3951000">So</span> <span m="3951320">as</span> <span m="3951550">m</span> <span m="3952480">goes</span> <span m="3952800">to</span> <span m="3952940">infinity,</span> <span m="3955660">the</span> <span m="3955840">expected</span> <span m="3956450">time</span> <span m="3956790">to</span> <span m="3956860">live</span> <span m="3957140">here</span> <span m="3958120">is</span> <span m="3958310">tilde</span> <span m="3959840">n</span> <span m="3960240">over</span> <span m="3960480">1</span> <span m="3960660">minus</span> <span m="3960970">2p.</span> <span m="3962060">If</span> <span m="3962150">you've</span> <span m="3962290">got</span> <span m="3962520">n</span> <span m="3962730">dollars,</span> <span m="3964330">losing</span> <span m="3964860">1</span> <span m="3965010">minus</span> <span m="3965320">2p</span> <span m="3965730">every</span> <span m="3965940">time,</span> <span m="3967330">then</span> <span m="3967540">you</span> <span m="3967660">last</span> <span m="3968030">for</span> <span m="3968150">n</span> <span m="3968320">over</span> <span m="3968490">1</span> <span m="3968640">minus</span> <span m="3968890">2p</span> <span m="3969220">steps.</span> </p>
<p><span m="3971650">OK,</span> <span m="3974100">now,</span> <span m="3974630">actually,</span> <span m="3975760">what</span> <span m="3976070">situation</span> <span m="3976370">in</span> <span m="3976670">words</span> <span m="3977180">does</span> <span m="3977580">m</span> <span m="3977930">going</span> <span m="3978230">to</span> <span m="3978350">infinity</span> <span m="3978870">mean?</span> <span m="3979900">Say</span> <span m="3980110">I</span> <span m="3980210">set</span> <span m="3980540">m</span> <span m="3980750">to</span> <span m="3980840">be</span> <span m="3980950">infinity?</span> <span m="3983200">What</span> <span m="3984350">is</span> <span m="3984450">that</span> <span m="3984610">kind</span> <span m="3984780">of</span> <span m="3984890">game</span> <span m="3985310">if</span> <span m="3985740">m</span> <span m="3986010">is infinity?</span> <span m="3986810">How</span> <span m="3987000">long</span> <span m="3987240">am I</span> <span m="3987360">playing</span> <span m="3987635">now?</span> <span m="3988120">Yeah.</span> </p>
<p><span m="3989140">AUDIENCE: Now you're</span> <span m="3989430">playing</span> <span m="3989720">for</span> <span m="3989900">as</span> <span m="3990160">long</span> <span m="3990420">as</span> <span m="3990520">it</span> <span m="3990670">takes</span> <span m="3991020">you</span> <span m="3991320">to lose</span> <span m="3991610">all of</span> <span m="3991870">your</span> <span m="3992240">money.</span> </p>
<p><span m="3992760">PROFESSOR: Yes,</span> <span m="3994240">because</span> <span m="3994620">there</span> <span m="3994880">is no</span> <span m="3995090">stopping</span> <span m="3995590">condition</span> <span m="3995930">up</span> <span m="3996070">here--</span> <span m="3996200">going</span> <span m="3996590">home</span> <span m="3996630">happy.</span> <span m="3997380">I'm</span> <span m="3997520">going</span> <span m="3997630">to</span> <span m="3997700">play</span> <span m="3998810">forever</span> <span m="3999470">or</span> <span m="3999670">until</span> <span m="4000670">I</span> <span m="4000820">lose</span> <span m="4001090">everything.</span> <span m="4003060">And</span> <span m="4003210">this</span> <span m="4003410">says</span> <span m="4003810">how</span> <span m="4004020">long</span> <span m="4004320">you</span> <span m="4004450">expect</span> <span m="4004870">to</span> <span m="4005240">play.</span> <span m="4007460">It's</span> <span m="4007610">a</span> <span m="4007660">little</span> <span m="4007890">less</span> <span m="4008150">than</span> <span m="4008280">n</span> <span m="4008410">over</span> <span m="4008540">1</span> <span m="4008730">minus</span> <span m="4009240">2p.</span> </p>
<p><span m="4011220">So</span> <span m="4011380">if</span> <span m="4011440">you</span> <span m="4011650">play</span> <span m="4011980">until</span> <span m="4012190">you</span> <span m="4012330">go</span> <span m="4012490">broke,</span> <span m="4013900">that's</span> <span m="4014120">how</span> <span m="4014230">long</span> <span m="4014430">you</span> <span m="4014570">expect</span> <span m="4014980">to</span> <span m="4015070">play.</span> <span m="4019950">So</span> <span m="4020500">that</span> <span m="4021270">sort</span> <span m="4021490">of</span> <span m="4021550">makes</span> <span m="4021770">sense</span> <span m="4022570">in</span> <span m="4022690">that</span> <span m="4022870">scenario.</span> <span m="4023200">That's</span> <span m="4023410">not</span> <span m="4023570">one</span> <span m="4023740">where it</span> <span m="4023970">surprises</span> <span m="4024390">you by</span> <span m="4024660">intuition.</span> </p>
<p><span m="4026570">It</span> <span m="4026670">is</span> <span m="4026840">interesting</span> <span m="4027190">to</span> <span m="4027260">consider</span> <span m="4027630">the</span> <span m="4027770">case</span> <span m="4028090">of</span> <span m="4028190">a</span> <span m="4028240">fair</span> <span m="4028590">game.</span> <span m="4028980">Because</span> <span m="4029140">there's</span> <span m="4029310">something</span> <span m="4029920">that's</span> <span m="4030140">non-intuitive</span> <span m="4030820">that</span> <span m="4030910">happens</span> <span m="4031330">there.</span> <span m="4032570">So</span> <span m="4032600">in</span> <span m="4032680">a</span> <span m="4032730">fair</span> <span m="4033060">game,</span> <span m="4033880">p</span> <span m="4034090">is</span> <span m="4034220">1/2.</span> </p>
<p><span m="4037350">Now,</span> <span m="4037440">if</span> <span m="4037520">I</span> <span m="4037590">plug in</span> <span m="4038010">1/2</span> <span m="4038430">here,</span> <span m="4040660">well,</span> <span m="4041860">I</span> <span m="4041980">divide</span> <span m="4042350">by</span> <span m="4042470">0.</span> <span m="4044280">I</span> <span m="4044440">expect</span> <span m="4044870">to</span> <span m="4044960">play</span> <span m="4046050">forever.</span> <span m="4047730">That's</span> <span m="4047980">not</span> <span m="4048110">a</span> <span m="4048150">good</span> <span m="4048360">way</span> <span m="4048500">to</span> <span m="4048590">do</span> <span m="4048720">the</span> <span m="4048850">analysis,</span> <span m="4049500">that</span> <span m="4049670">you</span> <span m="4050130">get</span> <span m="4050240">to</span> <span m="4050320">a</span> <span m="4050360">divide</span> <span m="4050620">by</span> <span m="4050710">0.</span> </p>
<p><span m="4050950">Let's</span> <span m="4051170">actually</span> <span m="4051420">go</span> <span m="4051580">back</span> <span m="4052680">and</span> <span m="4052970">look</span> <span m="4053170">at</span> <span m="4053280">this</span> <span m="4053580">for</span> <span m="4053650">the</span> <span m="4053760">case</span> <span m="4054060">when</span> <span m="4054210">p</span> <span m="4054430">is</span> <span m="4054610">1/2.</span> <span m="4057860">And</span> <span m="4058010">see</span> <span m="4058170">what</span> <span m="4058290">happens</span> <span m="4058660">in</span> <span m="4058740">a</span> <span m="4058800">fair</span> <span m="4059090">game--</span> <span m="4059310">how</span> <span m="4059500">long</span> <span m="4059720">you</span> <span m="4059840">expect</span> <span m="4060200">to</span> <span m="4060290">play</span> <span m="4060500">in</span> <span m="4060580">a</span> <span m="4060630">fair</span> <span m="4060870">game.</span> <span m="4063250">Then</span> <span m="4063460">the</span> <span m="4063550">homogeneous</span> <span m="4064240">solution</span> <span m="4067660">is</span> <span m="4067800">the</span> <span m="4067900">simple</span> <span m="4068220">case.</span> </p>
<p><span m="4068500">E</span> <span m="4068780">is</span> <span m="4069120">A n</span> <span m="4069540">plus</span> <span m="4069830">B.</span> <span m="4070620">You</span> <span m="4070700">have</span> <span m="4070840">a</span> <span m="4070890">double</span> <span m="4071200">root</span> <span m="4071460">at 1,</span> <span m="4074270">which</span> <span m="4074510">we</span> <span m="4074600">don't</span> <span m="4074760">have</span> <span m="4074860">to</span> <span m="4074970">worry</span> <span m="4075110">about</span> <span m="4075310">1 to</span> <span m="4075530">the</span> <span m="4075680">n.</span> <span m="4077340">When</span> <span m="4077440">you</span> <span m="4077520">do</span> <span m="4077630">your</span> <span m="4077790">particular</span> <span m="4078310">solution,</span> <span m="4083550">you'll</span> <span m="4083740">try</span> <span m="4085490">a</span> <span m="4085550">single</span> <span m="4085910">scalar,</span> <span m="4087480">and</span> <span m="4087660">it</span> <span m="4087790">fails.</span> <span m="4088620">I'll</span> <span m="4089160">use</span> <span m="4089320">lowercase</span> <span m="4089780">a--</span> <span m="4090020">fails.</span> </p>
<p><span m="4091730">You</span> <span m="4091930">will</span> <span m="4092050">then</span> <span m="4092270">try</span> <span m="4094110">a</span> <span m="4094220">degree</span> <span m="4094510">one</span> <span m="4094740">polynomial,</span> <span m="4096479">and</span> <span m="4096620">that</span> <span m="4096790">will</span> <span m="4096920">fail.</span> <span m="4098460">What</span> <span m="4098580">are</span> <span m="4098609">you</span> <span m="4098660">going</span> <span m="4098750">to</span> <span m="4098810">try</span> <span m="4099040">next?</span> <span m="4101010">Second-degree</span> <span m="4101640">polynomial,</span> <span m="4105770">and</span> <span m="4105899">that</span> <span m="4106090">will</span> <span m="4106220">work.</span> </p>
<p><span m="4108210">OK,</span> <span m="4108710">and the</span> <span m="4109200">answer</span> <span m="4109690">you</span> <span m="4110180">get</span> <span m="4113610">when you</span> <span m="4114100">do that</span> <span m="4119510">is</span> <span m="4119920">that--</span> <span m="4120899">I'll</span> <span m="4121109">put</span> <span m="4121279">the</span> <span m="4121410">answer</span> <span m="4121689">here.</span> <span m="4121960">It turns</span> <span m="4122250">out</span> <span m="4122410">that</span> <span m="4122569">a</span> <span m="4122969">is</span> <span m="4123770">minus</span> <span m="4124240">1</span> <span m="4125170">and</span> <span m="4125340">b</span> <span m="4125560">and</span> <span m="4125689">c</span> <span m="4125960">can</span> <span m="4126090">be</span> <span m="4126180">0.</span> <span m="4127450">So</span> <span m="4127649">it's</span> <span m="4127779">just</span> <span m="4128010">going</span> <span m="4128120">to</span> <span m="4128180">be</span> <span m="4128470">minus</span> <span m="4128950">n</span> <span m="4129090">squared</span> <span m="4129470">for</span> <span m="4129540">the</span> <span m="4129620">particular</span> <span m="4130069">solution.</span> <span m="4131660">That</span> <span m="4131899">means</span> <span m="4132290">your</span> <span m="4132410">general</span> <span m="4132800">solution</span> <span m="4133640">is</span> <span m="4135870">A</span> <span m="4136010">n</span> <span m="4136310">plus</span> <span m="4136660">B</span> <span m="4137470">minus</span> <span m="4137859">n</span> <span m="4138069">squared.</span> </p>
<p><span m="4140319">Now</span> <span m="4140420">you</span> <span m="4140520">do</span> <span m="4140620">your</span> <span m="4140649">boundary</span> <span m="4141069">condition.</span> <span m="4146660">You</span> <span m="4146680">have</span> <span m="4147090">E</span> <span m="4147310">0</span> <span m="4148060">is</span> <span m="4148330">0.</span> <span m="4150180">Plug in</span> <span m="4150569">0</span> <span m="4150870">for</span> <span m="4151100">n.</span> <span m="4151359">That's</span> <span m="4151609">equal</span> <span m="4151810">to</span> <span m="4151890">B.</span> <span m="4152880">So</span> <span m="4153069">B</span> <span m="4153290">is</span> <span m="4153430">0.</span> <span m="4153720">That's</span> <span m="4153990">nice.</span> </p>
<p><span m="4155620">E T</span> <span m="4157370">is</span> <span m="4158109">0.</span> <span m="4159180">And</span> <span m="4159270">I</span> <span m="4159330">plug in</span> <span m="4159760">T</span> <span m="4160060">here,</span> <span m="4160270">I</span> <span m="4160300">get</span> <span m="4160529">AT,</span> <span m="4161872">B</span> <span m="4162370">is</span> <span m="4162490">0</span> <span m="4162729">minus</span> <span m="4163090">T</span> <span m="4163340">squared.</span> <span m="4165029">So</span> <span m="4165430">I</span> <span m="4165550">solve</span> <span m="4166160">for</span> <span m="4166750">A</span> <span m="4167859">here.</span> </p>
<p><span m="4168160">That</span> <span m="4168450">means</span> <span m="4168779">that</span> <span m="4171250">A</span> <span m="4171520">equals</span> <span m="4171880">T.</span> <span m="4173756">AT</span> <span m="4174229">squared</span> <span m="4174490">minus</span> <span m="4174760">T</span> <span m="4174870">squared</span> <span m="4175090">is</span> <span m="4175189">0.</span> <span m="4175550">A has</span> <span m="4176020">to</span> <span m="4176109">be</span> <span m="4176260">T.</span> </p>
<p><span m="4178439">So</span> <span m="4178700">that</span> <span m="4178950">means</span> <span m="4179260">that</span> <span m="4182410">E n</span> <span m="4184740">is</span> <span m="4184930">Tn</span> <span m="4186700">minus</span> <span m="4187149">n</span> <span m="4187380">squared.</span> <span m="4189960">Now,</span> <span m="4190060">T</span> <span m="4191290">is</span> <span m="4191430">the</span> <span m="4191550">upper</span> <span m="4191720">bound.</span> <span m="4192090">It's</span> <span m="4192170">just</span> <span m="4192399">n</span> <span m="4192569">plus</span> <span m="4192840">m.</span> <span m="4196800">n</span> <span m="4197050">plus</span> <span m="4197340">m</span> <span m="4198340">times</span> <span m="4198630">n</span> <span m="4198980">minus</span> <span m="4199300">n</span> <span m="4199430">squared--</span> <span m="4200880">this</span> <span m="4201090">gets</span> <span m="4201290">really</span> <span m="4201650">simple.</span> </p>
<p><span m="4201925">The</span> <span m="4202200">m</span> <span m="4202370">squared</span> <span m="4202740">cancels.</span> <span m="4203190">I</span> <span m="4203240">just</span> <span m="4203500">get</span> <span m="4203650">n</span> <span m="4203800">out.</span> <span m="4206470">That</span> <span m="4206660">says</span> <span m="4206860">if</span> <span m="4206940">you're</span> <span m="4207050">playing</span> <span m="4207390">a</span> <span m="4207440">fair</span> <span m="4207790">game,</span> <span m="4208380">until</span> <span m="4208650">you</span> <span m="4208820">win</span> <span m="4209100">m</span> <span m="4209440">or</span> <span m="4209540">lose</span> <span m="4209850">n,</span> <span m="4210860">you</span> <span m="4211020">expect</span> <span m="4211450">to</span> <span m="4211530">play</span> <span m="4211890">for</span> <span m="4212100">nm</span> <span m="4212340">steps,</span> <span m="4214160">which</span> <span m="4214400">is</span> <span m="4214510">really</span> <span m="4214740">nice.</span> <span m="4215680">This</span> <span m="4215790">is</span> <span m="4215900">p</span> <span m="4216170">is</span> <span m="4216300">1/2--</span> <span m="4218760">very</span> <span m="4219020">clean.</span> </p>
<p><span m="4221100">Now,</span> <span m="4222930">if</span> <span m="4223070">you</span> <span m="4223190">let</span> <span m="4223370">m</span> <span m="4223630">equal</span> <span m="4223920">to</span> <span m="4224010">infinity,</span> <span m="4225940">you're</span> <span m="4226100">going</span> <span m="4226220">to</span> <span m="4226690">expect</span> <span m="4227340">to</span> <span m="4227420">play</span> <span m="4227760">forever.</span> <span m="4229680">So</span> <span m="4229800">with</span> <span m="4229840">a</span> <span m="4229930">fair</span> <span m="4230320">game,</span> <span m="4231300">if</span> <span m="4231390">you</span> <span m="4231510">play</span> <span m="4231800">until</span> <span m="4232030">you're</span> <span m="4232150">broke,</span> <span m="4233680">the</span> <span m="4233780">expected</span> <span m="4234270">number</span> <span m="4234550">of</span> <span m="4234650">bets</span> <span m="4235390">is</span> <span m="4235570">infinite.</span> <span m="4237220">That's</span> <span m="4237470">nice.</span> <span m="4239510">You</span> <span m="4240550">can play</span> <span m="4240960">forever</span> <span m="4241400">is</span> <span m="4241490">the</span> <span m="4241580">expectation.</span> </p>
<p><span m="4246160">Now,</span> <span m="4248060">here's</span> <span m="4248480">the</span> <span m="4248600">weird</span> <span m="4248930">thing.</span> <span m="4250430">If</span> <span m="4250570">you</span> <span m="4250730">expect</span> <span m="4251240">to</span> <span m="4251340">play</span> <span m="4251670">forever,</span> <span m="4253080">does</span> <span m="4253220">that</span> <span m="4253420">mean</span> <span m="4253670">you're</span> <span m="4253790">not</span> <span m="4254130">likely</span> <span m="4254300">to go</span> <span m="4254775">home</span> <span m="4255250">broke?</span> <span m="4258580">You</span> <span m="4258750">expect</span> <span m="4259290">to</span> <span m="4259370">play</span> <span m="4261590">forever.</span> <span m="4262410">And as</span> <span m="4262580">long</span> <span m="4262780">as</span> <span m="4262840">you're</span> <span m="4262930">playing,</span> <span m="4263270">you're</span> <span m="4263350">not</span> <span m="4263500">going</span> <span m="4263710">home</span> <span m="4263900">broke.</span> </p>
<p><span m="4265980">Now,</span> <span m="4266050">there's</span> <span m="4266440">some</span> <span m="4266700">chance</span> <span m="4266960">of</span> <span m="4267040">going</span> <span m="4267220">home</span> <span m="4267480">broke,</span> <span m="4267850">because you</span> <span m="4267930">might</span> <span m="4268090">just</span> <span m="4268260">lose</span> <span m="4268440">every</span> <span m="4268640">bet--</span> <span m="4269620">not</span> <span m="4269840">likely.</span> <span m="4275930">Here's</span> <span m="4276150">the</span> <span m="4276250">weird</span> <span m="4276510">thing--</span> <span m="4277820">the</span> <span m="4277940">probability</span> <span m="4278580">you</span> <span m="4278680">go</span> <span m="4278880">home</span> <span m="4279130">broke</span> <span m="4279490">if</span> <span m="4279600">you</span> <span m="4279720">play</span> <span m="4280590">until</span> <span m="4280810">you</span> <span m="4280910">go</span> <span m="4281050">broke</span> <span m="4282040">is</span> <span m="4282240">1.</span> <span m="4284460">You</span> <span m="4284660">will</span> <span m="4285150">go</span> <span m="4285370">home</span> <span m="4285600">broke.</span> </p>
<p><span m="4286970">It's</span> <span m="4287230">just</span> <span m="4287500">that</span> <span m="4287610">it</span> <span m="4287740">takes</span> <span m="4288090">you</span> <span m="4289260">expected</span> <span m="4290140">infinite</span> <span m="4290520">amount</span> <span m="4290760">of</span> <span m="4290840">time</span> <span m="4291170">to</span> <span m="4291280">do</span> <span m="4291490">it--</span> <span m="4292930">sort</span> <span m="4293190">of</span> <span m="4293560">one</span> <span m="4293710">of</span> <span m="4293750">these</span> <span m="4293920">weird</span> <span m="4294270">things</span> <span m="4295580">in</span> <span m="4295730">a</span> <span m="4295780">fair</span> <span m="4296000">game.</span> <span m="4296350">So</span> <span m="4296440">here</span> <span m="4296620">we</span> <span m="4297590">proved</span> <span m="4298020">the</span> <span m="4298810">expected</span> <span m="4299290">number</span> <span m="4299550">bets</span> <span m="4299820">is</span> <span m="4299940">nm.</span> <span m="4300330">If</span> <span m="4300450">m</span> <span m="4300680">is</span> <span m="4300770">infinite,</span> <span m="4301370">that</span> <span m="4301530">becomes</span> <span m="4301820">an</span> <span m="4301910">infinite</span> <span m="4302220">number</span> <span m="4302460">of</span> <span m="4302530">bets.</span> </p>
<p><span m="4304420">One</span> <span m="4304710">more</span> <span m="4304940">theorem</span> <span m="4305340">here--</span> <span m="4307190">this</span> <span m="4307260">one's</span> <span m="4307430">a</span> <span m="4307480">little</span> <span m="4307700">surprising.</span> <span m="4312330">This</span> <span m="4312600">theorem is</span> <span m="4312730">called</span> <span m="4313000">Quit</span> <span m="4313270">While</span> <span m="4313450">You're</span> <span m="4313680">Ahead.</span> <span m="4326570">If</span> <span m="4326780">you</span> <span m="4326950">start</span> <span m="4331050">with</span> <span m="4331340">n</span> <span m="4331610">dollars,</span> <span m="4334560">and</span> <span m="4335000">it's</span> <span m="4335170">a</span> <span m="4335230">fair</span> <span m="4335620">game,</span> <span m="4338700">and</span> <span m="4339030">you</span> <span m="4339160">play</span> <span m="4342990">until</span> <span m="4343290">you</span> <span m="4343430">go</span> <span m="4343610">broke,</span> <span m="4352930">then</span> <span m="4353700">the</span> <span m="4353800">probability</span> <span m="4354870">that</span> <span m="4355010">you</span> <span m="4355150">do</span> <span m="4355420">go</span> <span m="4355610">broke,</span> <span m="4358540">as</span> <span m="4358730">opposed</span> <span m="4359100">to</span> <span m="4359170">playing</span> <span m="4359500">forever,</span> <span m="4360750">is</span> <span m="4361020">1.</span> </p>
<p><span m="4361730">It's</span> <span m="4361910">a</span> <span m="4361980">certainty.</span> <span m="4363430">You'll</span> <span m="4363590">go</span> <span m="4363750">broke,</span> <span m="4364550">even</span> <span m="4364810">though</span> <span m="4366240">you</span> <span m="4366360">expect</span> <span m="4366640">it to</span> <span m="4366730">take</span> <span m="4367140">an</span> <span m="4367260">infinite</span> <span m="4367540">amount</span> <span m="4367760">of</span> <span m="4367830">time.</span> <span m="4369570">All right,</span> <span m="4369610">so let's</span> <span m="4370098">prove</span> <span m="4370586">that.</span> </p>
<p></p>
<p><span m="4386700">OK,</span> <span m="4386840">the</span> <span m="4386970">proof</span> <span m="4387330">is</span> <span m="4387470">by</span> <span m="4387590">contradiction.</span> <span m="4395330">Assume</span> <span m="4395910">it's</span> <span m="4396090">not</span> <span m="4396580">true.</span> <span m="4398600">And</span> <span m="4398780">that</span> <span m="4398980">means</span> <span m="4399410">that</span> <span m="4399880">you're</span> <span m="4400090">assuming</span> <span m="4403220">that</span> <span m="4403400">there</span> <span m="4403580">exists</span> <span m="4404140">some</span> <span m="4404470">number</span> <span m="4404770">of</span> <span m="4404860">dollars</span> <span m="4406080">that</span> <span m="4406100">you</span> <span m="4406240">can</span> <span m="4406360">start</span> <span m="4406650">with,</span> <span m="4408270">and</span> <span m="4408670">some</span> <span m="4409250">epsilon</span> <span m="4409920">bigger</span> <span m="4410170">than</span> <span m="4410340">0,</span> <span m="4412880">such</span> <span m="4413260">that</span> <span m="4417650">the</span> <span m="4417760">probability</span> <span m="4419600">that</span> <span m="4419760">you</span> <span m="4420400">lose</span> <span m="4420880">the</span> <span m="4421010">n</span> <span m="4421190">dollars--</span> <span m="4422400">in</span> <span m="4423070">which</span> <span m="4423260">case</span> <span m="4423440">you're</span> <span m="4423530">going</span> <span m="4423720">home</span> <span m="4423900">broke--</span> <span m="4427770">let</span> <span m="4427840">me</span> <span m="4428950">write</span> <span m="4429190">the</span> <span m="4429320">probability</span> <span m="4429800">you go</span> <span m="4429990">broke--</span> <span m="4436310">is</span> <span m="4436840">at</span> <span m="4436980">most</span> <span m="4437600">1</span> <span m="4437800">minus</span> <span m="4438140">epsilon.</span> </p>
<p><span m="4440410">In other</span> <span m="4440600">words,</span> <span m="4440740">if</span> <span m="4440840">the</span> <span m="4440920">theorem</span> <span m="4441170">is</span> <span m="4441250">not</span> <span m="4441500">true,</span> <span m="4441800">there's</span> <span m="4442060">some</span> <span m="4442270">amount</span> <span m="4442510">of</span> <span m="4442560">money</span> <span m="4442770">you</span> <span m="4442850">can</span> <span m="4442980">start</span> <span m="4443280">with</span> <span m="4444520">such</span> <span m="4444860">that the</span> <span m="4444940">chance</span> <span m="4445190">you</span> <span m="4445280">go</span> <span m="4445410">broke</span> <span m="4445700">is</span> <span m="4446530">less</span> <span m="4446810">than</span> <span m="4447220">1--</span> <span m="4448580">less</span> <span m="4448660">than</span> <span m="4448850">1</span> <span m="4449000">minus</span> <span m="4449280">epsilon.</span> <span m="4452150">OK,</span> <span m="4452480">now</span> <span m="4452620">that</span> <span m="4452910">means</span> <span m="4453260">that</span> <span m="4453440">for</span> <span m="4453650">all</span> <span m="4454040">m,</span> <span m="4456550">where</span> <span m="4456710">you</span> <span m="4456820">might</span> <span m="4457120">possibly</span> <span m="4457590">stop</span> <span m="4458030">but</span> <span m="4458130">you're</span> <span m="4458230">not</span> <span m="4458430">going</span> <span m="4458670">to,</span> <span m="4459930">the</span> <span m="4460030">probability</span> <span m="4460680">you</span> <span m="4460810">lose</span> <span m="4462470">n</span> <span m="4463870">before</span> <span m="4464380">you</span> <span m="4464530">win</span> <span m="4465120">m</span> <span m="4469910">is</span> <span m="4470160">at</span> <span m="4470260">most</span> <span m="4470610">1</span> <span m="4470760">minus</span> <span m="4471070">epsilon.</span> <span m="4475700">Because</span> <span m="4476380">we're</span> <span m="4476560">saying</span> <span m="4476840">the probability</span> <span m="4477240">you</span> <span m="4477380">lose</span> <span m="4477710">n</span> <span m="4477980">no</span> <span m="4478040">matter</span> <span m="4478280">what</span> <span m="4478550">is</span> <span m="4478730">at</span> <span m="4478830">most</span> <span m="4479070">that.</span> <span m="4479360">So</span> <span m="4479530">it's</span> <span m="4479670">certainly</span> <span m="4480860">less</span> <span m="4481090">than</span> <span m="4481170">1</span> <span m="4481290">minus</span> <span m="4481680">epsilon that</span> <span m="4481960">you</span> <span m="4482350">lose</span> <span m="4482770">n</span> <span m="4483030">before</span> <span m="4483330">you</span> <span m="4483450">win</span> <span m="4483790">m</span> <span m="4483970">dollars.</span> </p>
<p><span m="4486670">And</span> <span m="4486810">we</span> <span m="4486900">know</span> <span m="4487080">what</span> <span m="4487210">that</span> <span m="4487380">probability</span> <span m="4487960">is.</span> <span m="4488880">This</span> <span m="4489070">probability</span> <span m="4489580">is</span> <span m="4489710">just</span> <span m="4490060">m</span> <span m="4490450">over</span> <span m="4490720">n</span> <span m="4490900">plus</span> <span m="4491160">m.</span> <span m="4491770">We</span> <span m="4491920">proved</span> <span m="4492170">that</span> <span m="4492500">earlier.</span> <span m="4494700">So</span> <span m="4494850">that</span> <span m="4495090">has</span> <span m="4495350">to</span> <span m="4495430">be</span> <span m="4495530">less</span> <span m="4495780">than</span> <span m="4495890">1</span> <span m="4496040">minus</span> <span m="4496350">epsilon</span> <span m="4496840">for</span> <span m="4497020">all</span> <span m="4497360">m.</span> </p>
<p><span m="4499360">And</span> <span m="4499480">now</span> <span m="4499690">I</span> <span m="4499790">just</span> <span m="4500000">multiply</span> <span m="4500490">through</span> <span m="4503090">for</span> <span m="4503220">all</span> <span m="4503530">m.</span> <span m="4505150">That</span> <span m="4505340">means</span> <span m="4505560">that</span> <span m="4505700">m</span> <span m="4506180">is</span> <span m="4506400">less</span> <span m="4506670">than</span> <span m="4506820">or</span> <span m="4506910">equal</span> <span m="4507160">to</span> <span m="4507320">1</span> <span m="4507570">minus</span> <span m="4507930">epsilon</span> <span m="4508670">n plus</span> <span m="4509143">m.</span> <span m="4512930">And</span> <span m="4513260">then</span> <span m="4513360">we'll</span> <span m="4513480">solve</span> <span m="4516600">that.</span> </p>
<p><span m="4525960">OK,</span> <span m="4526650">so</span> <span m="4528010">just</span> <span m="4528120">multiply</span> <span m="4528630">this</span> <span m="4528900">out.</span> <span m="4529630">So</span> <span m="4529830">for</span> <span m="4530030">all</span> <span m="4530280">m</span> <span m="4533490">less</span> <span m="4533590">than</span> <span m="4533730">or</span> <span m="4533840">equal</span> <span m="4534070">to</span> <span m="4534210">n</span> <span m="4534390">plus</span> <span m="4534700">m</span> <span m="4538480">minus</span> <span m="4538990">epsilon</span> <span m="4539260">n</span> <span m="4540180">minus</span> <span m="4540390">epsilon</span> <span m="4540630">m,</span> <span m="4541960">and</span> <span m="4542100">now</span> <span m="4542290">pull</span> <span m="4542580">the</span> <span m="4542750">m</span> <span m="4542950">terms</span> <span m="4543330">out</span> <span m="4543530">here,</span> <span m="4545360">I</span> <span m="4545500">get</span> <span m="4546450">for</span> <span m="4546630">all</span> <span m="4546970">m,</span> <span m="4548060">epsilon</span> <span m="4548530">m</span> <span m="4548980">is</span> <span m="4549210">less</span> <span m="4549430">than</span> <span m="4549640">or</span> <span m="4549730">equal</span> <span m="4549970">to</span> <span m="4550480">1</span> <span m="4550680">minus</span> <span m="4551040">epsilon</span> <span m="4551500">n.</span> <span m="4553020">That</span> <span m="4553180">means</span> <span m="4553420">for</span> <span m="4553650">all</span> <span m="4553860">m,</span> <span m="4554672">m</span> <span m="4555080">is</span> <span m="4555290">smaller</span> <span m="4555840">than</span> <span m="4556860">1</span> <span m="4557010">minus</span> <span m="4557340">epsilon</span> <span m="4557770">over</span> <span m="4557990">epsilon</span> <span m="4558910">times</span> <span m="4559320">n.</span> </p>
<p><span m="4561170">And</span> <span m="4561380">that</span> <span m="4561630">can't</span> <span m="4561930">be</span> <span m="4562060">true.</span> <span m="4563060">It's</span> <span m="4563230">not</span> <span m="4563510">true</span> <span m="4563650">the</span> <span m="4563780">for</span> <span m="4563960">all</span> <span m="4564210">m,</span> <span m="4564560">this</span> <span m="4564750">is</span> <span m="4564890">less</span> <span m="4565150">than</span> <span m="4565260">that,</span> <span m="4566270">because</span> <span m="4566440">these</span> <span m="4566620">are</span> <span m="4566700">fixed</span> <span m="4567040">values.</span> <span m="4569030">That's</span> <span m="4569100">a</span> <span m="4569150">contradiction.</span> </p>
<p><span m="4571790">All</span> <span m="4571880">right,</span> <span m="4572180">so</span> <span m="4572270">we</span> <span m="4572430">proved</span> <span m="4573840">that</span> <span m="4574960">if</span> <span m="4575140">you</span> <span m="4575320">keep</span> <span m="4575590">playing</span> <span m="4575850">until</span> <span m="4576040">you're</span> <span m="4576180">broke,</span> <span m="4577040">you</span> <span m="4577170">will</span> <span m="4577290">go</span> <span m="4577530">broke</span> <span m="4577940">with</span> <span m="4578100">probability</span> <span m="4578660">1.</span> <span m="4580030">So</span> <span m="4580260">even</span> <span m="4580540">if</span> <span m="4580640">you're</span> <span m="4580740">playing</span> <span m="4581120">a</span> <span m="4581170">fair</span> <span m="4581490">game,</span> <span m="4583240">quit</span> <span m="4583770">while</span> <span m="4584010">you're</span> <span m="4584210">ahead.</span> <span m="4585060">Because</span> <span m="4585280">if</span> <span m="4585370">you</span> <span m="4585470">don't,</span> <span m="4586990">you're</span> <span m="4587130">going</span> <span m="4587230">to</span> <span m="4587280">go</span> <span m="4587960">broke.</span> <span m="4588970">The</span> <span m="4589080">swings</span> <span m="4589600">will</span> <span m="4589790">eventually</span> <span m="4590280">catch</span> <span m="4590560">up</span> <span m="4590700">with</span> <span m="4590840">you.</span> <span m="4591500">So</span> <span m="4591620">if</span> <span m="4591700">we</span> <span m="4591770">draw</span> <span m="4592080">the</span> <span m="4592190">graph</span> <span m="4592550">here,</span> <span m="4593860">we'll</span> <span m="4593940">see</span> <span m="4594150">why</span> <span m="4594310">that's</span> <span m="4594670">true.</span> </p>
<p></p>
<p><span m="4611810">All</span> <span m="4611900">right,</span> <span m="4612040">if</span> <span m="4612150">I</span> <span m="4612200">have</span> <span m="4612360">time</span> <span m="4612760">going</span> <span m="4613060">this</span> <span m="4613260">way,</span> <span m="4615660">and</span> <span m="4615840">I</span> <span m="4615910">start</span> <span m="4616300">with</span> <span m="4616440">n</span> <span m="4616610">dollars,</span> <span m="4620240">my</span> <span m="4620780">baseline</span> <span m="4621430">is</span> <span m="4621550">here.</span> <span m="4621750">The</span> <span m="4621890">drift</span> <span m="4622240">is</span> <span m="4622350">0.</span> <span m="4624000">I'm</span> <span m="4624170">going</span> <span m="4624280">to</span> <span m="4624350">have</span> <span m="4624540">swings.</span> </p>
<p><span m="4627480">I</span> <span m="4627570">might</span> <span m="4627750">have</span> <span m="4627890">some</span> <span m="4628050">really</span> <span m="4628400">big,</span> <span m="4628690">high</span> <span m="4628970">swings,</span> <span m="4629510">but it</span> <span m="4629570">doesn't</span> <span m="4629850">matter,</span> <span m="4630680">because</span> <span m="4630850">eventually</span> <span m="4632550">I'm</span> <span m="4632700">going</span> <span m="4632840">to</span> <span m="4632890">get</span> <span m="4633030">a</span> <span m="4633090">really</span> <span m="4633320">bad</span> <span m="4633650">swing,</span> <span m="4634720">and</span> <span m="4634870">I'm</span> <span m="4634940">going</span> <span m="4635040">to</span> <span m="4635420">go</span> <span m="4635600">broke.</span> <span m="4639450">Now,</span> <span m="4639940">if</span> <span m="4640130">you</span> <span m="4640360">ever</span> <span m="4640400">play a</span> <span m="4640820">game</span> <span m="4641040">where</span> <span m="4641190">you're</span> <span m="4641370">likely</span> <span m="4641770">to</span> <span m="4641840">be</span> <span m="4641970">winning</span> <span m="4642350">each</span> <span m="4642550">time,</span> <span m="4644080">and the</span> <span m="4644230">drift</span> <span m="4644550">goes</span> <span m="4644800">up,</span> <span m="4645960">that's</span> <span m="4646210">a</span> <span m="4646270">good</span> <span m="4646440">game</span> <span m="4646640">to</span> <span m="4646720">play,</span> <span m="4647360">obviously.</span> <span m="4647635">It</span> <span m="4647910">just</span> <span m="4648160">keeps</span> <span m="4648640">getting</span> <span m="4648900">better.</span> <span m="4649620">And</span> <span m="4649700">that's</span> <span m="4649880">a</span> <span m="4649930">whole</span> <span m="4650140">math</span> <span m="4650460">change</span> <span m="4650820">there.</span> </p>
<p><span m="4654410">So</span> <span m="4654710">that's</span> <span m="4655000">it.</span> <span m="4656380">Remember,</span> <span m="4656890">we</span> <span m="4657000">have</span> <span m="4657190">the</span> <span m="4657310">ice</span> <span m="4657470">cream</span> <span m="4657640">study</span> <span m="4657920">session</span> <span m="4658330">Monday.</span> <span m="4659030">So</span> <span m="4659630">come</span> <span m="4659850">to</span> <span m="4659910">that</span> <span m="4660160">if</span> <span m="4660260">you'd</span> <span m="4660410">like.</span> <span m="4661790">And</span> <span m="4661970">definitely</span> <span m="4662440">come</span> <span m="4662820">to</span> <span m="4662980">the</span> <span m="4663080">final</span> <span m="4663720">on</span> <span m="4663950">Tuesday.</span> <span m="4665470">And</span> <span m="4666130">thanks</span> <span m="4666360">for</span> <span m="4666420">your</span> <span m="4666570">hard</span> <span m="4666810">work,</span> <span m="4667090">and</span> <span m="4667270">being</span> <span m="4667440">such</span> <span m="4667610">a</span> <span m="4667670">great</span> <span m="4667970">class</span> <span m="4668270">this year.</span> </p>
<p><span m="4669170">[APPLAUSE]</span> </p>
<p></p>
</div>
        <div id="vid_related" itemprop="description" class="tabContent hide">
<h2 class="subhead">Free Downloads</h2>
<h3 class="subsubhead">Video</h3>
<ul>
<li>iTunes U (<a href="http://itunes.apple.com/us/itunes-u/lecture-25-random-walks/id503873536?i=110644958">MP4 - 170MB</a>)</li>
<li>Internet Archive (<a href="http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec25_300k.mp4">MP4 - 170MB</a>)</li>
</ul>
<br>
<h3 class="subsubhead">Caption</h3>
<ul>
<li>English-US (<a href="../../../contents/video-lectures/lecture-25-random-walks/56iFMY8QW2k.srt">SRT</a>)</li>
</ul>
</div>
    
   </div>  




      					 
        <div class="" id="parent-fieldname-bottom_html_area">
            
            
        </div>
    
               </main><!--Course_inner_media tag close -->
           		</div>
<!--Course_wrapper tag close -->
            </div>
<!--left tag close -->
            <aside id="right">
                <!--Begin Right Portion -->
                    <div>
    



</div>

                	<div>
    



</div>


        <div class="" id="parent-fieldname-rsi_top_html_area">
            
            
        </div>
    

<!-- RSI google ad space-->



<div id="google_ads">    
    <script async="async" src="https://www.googletagservices.com/tag/js/gpt.js"></script>
    <script type="text/javascript">var googletag = googletag || {}; googletag.cmd = googletag.cmd || [];</script>
    <script type="text/javascript">
googletag.cmd.push(function() {googletag.defineSlot('/1064917/VIDEO_INDIVIDUAL_SLOT_A_DL', [[300, 250], [300, 300], [180, 200], [180, 150], [160, 600]], 'VIDEO_INDIVIDUAL_SLOT_A_DL').addService(googletag.pubads());googletag.defineSlot('/1064917/VIDEO_INDIVIDUAL_SLOT_B_DL', [[300, 250], [300, 300], [180, 200], [180, 150], [160, 600]], 'VIDEO_INDIVIDUAL_SLOT_B_DL').addService(googletag.pubads());googletag.defineSlot('/1064917/VIDEO_INDIVIDUAL_SLOT_C_DL', [[300, 250], [300, 300], [180, 200], [180, 150], [160, 600]], 'VIDEO_INDIVIDUAL_SLOT_C_DL').addService(googletag.pubads());
googletag.pubads().enableSingleRequest();
 googletag.enableServices();
});</script>
    <script language="javascript" type="text/javascript">
googletag.cmd.push(function() {googletag.pubads().set("TYPE","HOUSE");googletag.pubads().set("DEPARTMENT","6");googletag.pubads().set("CRS_BEG2","04");googletag.pubads().set("CRS_END","2J");googletag.pubads().set("SESSION","F");googletag.pubads().set("YEAR","10");})
</script>
    
    <div id="VIDEO_INDIVIDUAL_SLOT_A_DL">
    	<script>googletag.cmd.push(function() { googletag.display('VIDEO_INDIVIDUAL_SLOT_A_DL'); });</script>
    </div>
    <div id="VIDEO_INDIVIDUAL_SLOT_B_DL">
    	<script>googletag.cmd.push(function() { googletag.display('VIDEO_INDIVIDUAL_SLOT_B_DL'); });</script>
    </div>
    <div id="VIDEO_INDIVIDUAL_SLOT_C_DL">
    	<script>googletag.cmd.push(function() { googletag.display('VIDEO_INDIVIDUAL_SLOT_C_DL'); });</script>
    </div>
</div>

<!-- End RSI ads--> 


<div>
    



</div>

            </aside><!--Right div close -->
            <div class="clear"></div>
        </div>
<!--grid tag close -->
      </div>
		
		<footer id="bottom">
			<div id="grid">
				
<div id="portletwrapper-6f63772e626f74746f6d706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d666f6f746572" class="portletWrapper kssattr-portlethash-6f63772e626f74746f6d706f72746c65746d616e616765720a636f6e746578740a2f506c6f6e650a736974652d666f6f746572">
<div class="portletStaticText portlet-static-site-footer"><div id="footer">
<nav aria-label="Footer">     <nav id="foot-c1" class="grid_2 alpha" aria-labelledby="f-find-courses">       <span class="footer" id="f-find-courses" aria-hidden="true">Find Courses</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/courses/find-by-topic/">Find by Topic</a></li>
    <li><a href="https://ocw.mit.edu/courses/find-by-number/">Find by Course Number</a></li>
    <li><a href="https://ocw.mit.edu/courses/find-by-department/">Find by Department</a></li>
    <li><a href="https://ocw.mit.edu/courses/new-courses/">New Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/most-visited-courses/">Most Visited Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/ocw-scholar/">OCW Scholar Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/audio-video-courses/">Audio/Video Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/online-textbooks/">Online Textbooks</a></li>
    <li><a href="https://ocw.mit.edu/courses/instructor-insights/">Instructor Insights</a></li>
    <li><a href="https://ocw.mit.edu/resources/">Supplemental Resources</a></li>
    <li><a href="https://ocw.mit.edu/courses/mitx-related-courseware/">MITx &amp; Related OCW Courses</a></li>
    <li><a href="https://ocw.mit.edu/courses/mit-open-learning-library/">MIT Open Learning Library</a></li>
    <li><a href="https://ocw.mit.edu/courses/translated-courses/">Translated Courses</a></li>
</ul>
</nav>
<div id="foot-c2" class="grid_2"><nav aria-labelledby="f-for-educators">         <span id="f-for-educators" class="footer" aria-hidden="true">For Educators</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://chalk-radio.simplecast.com/">Chalk Radio Podcast</a></li>
    <li><a href="https://ocw.mit.edu/educator/">OCW Educator Portal </a></li>
    <li><a href="https://ocw.mit.edu/courses/instructor-insights/">Instructor Insights by Department</a></li>
    <li><a href="https://openlearning.mit.edu/campus/digital-innovations/" aria-label="External Link: Residential Digital Innovations">Residential Digital Innovations </a></li>
    <li><a href="https://ocw.mit.edu/high-school/">OCW Highlights for High School</a></li>
    <li><a href="https://ocw.mit.edu/educator/additional-resources/">Additional Resources</a></li>
</ul>
</nav></div>
<nav class="grid_2" id="foot-c3" aria-labelledby="f-donate">       <span id="f-donate" class="footer" aria-hidden="true">Give Now</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/give/">Make a Donation</a></li>
    <li><a href="https://ocw.mit.edu/give/why-give/">Why Give?</a></li>
    <li><a href="https://ocw.mit.edu/give/our-supporters/">Our Supporters</a></li>
    <li><a href="https://ocw.mit.edu/give/other-ways-to-contribute/">Other Ways to Contribute</a></li>
    <li><a href="https://ocw.mit.edu/support/">Become a Corporate Sponsor</a></li>
</ul>
</nav>
<div class="grid_2" id="foot-c4">
<nav aria-labelledby="f-about">         <span id="f-about" class="footer" aria-hidden="true">About</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/about/">About OpenCourseWare</a></li>
    <li><a href="https://ocw.mit.edu/about/site-statistics/">Site Statistics</a></li>
    <li><a href="https://ocw.mit.edu/about/ocw-stories/">OCW Stories</a></li>
    <li><a href="https://ocw.mit.edu/about/newsletter/">Newsletter</a></li>
    <li><a href="https://www.ocw-openmatters.org/">Open Matters Blog</a></li>
</ul>
</nav><!--about-->       <nav aria-labelledby="f-tools">         <span id="f-tools" class="footer" aria-hidden="true">Tools</span>
<ul class="foot-bullet" role="presentation">
    <li><a href="https://ocw.mit.edu/help/">Help &amp; FAQs</a></li>
    <li><a href="https://ocw.mit.edu/about/contactus">Contact Us</a></li>
    <li><a href="https://accessibility.mit.edu/" target="_blank">Accessibility</a></li>
    <li><a href="https://ocw.mit.edu/help/site-map/">Site Map</a></li>
    <li><a href="../../../common/terms/index.htm">Privacy &amp; Terms of Use</a></li>
    <li><a href="https://ocw.mit.edu/help/rss/">RSS Feeds</a></li>
</ul>
</nav><!--tools-->
</div>
</nav> <aside style="min-height: 289px;" aria-labelledby="f-our-corporate-supporters" class="grid_4 omega" id="foot-c5">           <span aria-hidden="true" class="footer" id="f-our-corporate-supporters">Our Corporate Supporters</span>           <!-- HOME_CORP_LOGO_1 -->
<div id="div-gpt-ad-1388181177156-0" class="sponsors_google_ads_even"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-0'); });
            </script></div>
<!-- HOME_CORP_LOGO_2 -->
<div id="div-gpt-ad-1388181177156-1" class="sponsors_google_ads_odd"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-1'); });
            </script></div>
<!-- HOME_CORP_LOGO_3 -->
<div id="div-gpt-ad-1388181177156-2" class="sponsors_google_ads_even"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-2'); });
            </script></div>
<!-- HOME_CORP_LOGO_4 -->
<div id="div-gpt-ad-1388181177156-3" class="sponsors_google_ads_odd"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-3'); });
            </script></div>
<!-- HOME_CORP_LOGO_5 -->
<div id="div-gpt-ad-1388181177156-4" class="sponsors_google_ads_even"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-4'); });
              </script></div>
<!-- HOME_CORP_LOGO_6 -->
<div id="div-gpt-ad-1388181177156-5" class="sponsors_google_ads_odd"><script type="text/javascript">
              googletag.cmd.push(function() { googletag.display('div-gpt-ad-1388181177156-5'); });
              </script></div>
</aside>
<div class="grid_12 alpha omega" itemprop="publisher" itemscope="" itemtype="http://schema.org/CollegeOrUniversity">
<h4 class="footer" style="border-top: thin solid #d5c9ba; padding-top: 10px; margin-bottom: 10px;">About <span itemprop="name">MIT OpenCourseWare</span>
</h4>
<p style="color: #999; font-size: 1em; line-height: 1.5em; margin-top: 10px;" itemprop="description">MIT OpenCourseWare is an online publication of materials from over 2,500 MIT courses, freely sharing knowledge with learners and educators around the world. <a href="https://ocw.mit.edu/about/">Learn more »</a></p>
</div>
<div id="foot-copy" class="grid_12 alpha omega" style="border-top: none;">
<a href="http://web.mit.edu"><img src="../../../common/images/logo_mit.png" alt="Massachusetts Institute of Technology logo and name." style="width: 195; height: 44;"></a><a href="https://openlearning.mit.edu/"><img src="https://ocw.mit.edu/images/mitol_logo.png" alt="MIT Open Learning logo and name." style="width: 265; height: 50; vertical-align: top; padding-left:30px;"></a><a href="https://www.oeglobal.org/"><img src="https://ocw.mit.edu/images/oeg_logo.gif" alt="Open Education Consortium logo." style="width: 219px; height: 59px; vertical-align: top; padding-left:20px;"></a><a rel="license" itemprop="useRightsUrl" href="http://creativecommons.org/licenses/by-nc-sa/4.0/"><img src="../../../common/images/cc_by-nc-sa.png" alt="Creative Commons logo with terms BY-NC-SA." style="width: 126px; height: 44px; margin-right: 0; padding-left: 20px;"></a>
<p class="copyright">© 2001–2018<br>
Massachusetts Institute of Technology</p>
<p style="font-size: 0.9em; margin-bottom: 15px;">Your use of the MIT OpenCourseWare site and materials is subject to our <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="license">Creative Commons License</a> and other <a href="../../../common/terms/index.htm" rel="cc:morePermissions">terms of use</a>.</p>
</div>
</div></div>

</div>





                
			</div> <!-- bottom grid end -->
		</footer><!-- footer bottom end -->


   </body>
 </html>
